diff --git a/src/Psalm/Internal/Analyzer/Statements/Expression/Call/FunctionCallAnalyzer.php b/src/Psalm/Internal/Analyzer/Statements/Expression/Call/FunctionCallAnalyzer.php
index 508f504e6..367df0290 100644
--- a/src/Psalm/Internal/Analyzer/Statements/Expression/Call/FunctionCallAnalyzer.php
+++ b/src/Psalm/Internal/Analyzer/Statements/Expression/Call/FunctionCallAnalyzer.php
@@ -200,10 +200,39 @@ class FunctionCallAnalyzer extends \Psalm\Internal\Analyzer\Statements\Expressio
                 $stmt->inferredType = Type::getMixed();
             }
         } else {
-            $function_id = implode('\\', $stmt->name->parts);
+            $original_function_id = implode('\\', $stmt->name->parts);
 
-            $in_call_map = CallMap::inCallMap($function_id);
-            $is_stubbed = $codebase_functions->hasStubbedFunction($function_id);
+            if (!$stmt->name instanceof PhpParser\Node\Name\FullyQualified) {
+                $function_id = $codebase_functions->getFullyQualifiedFunctionNameFromString(
+                    $original_function_id,
+                    $statements_analyzer
+                );
+            } else {
+                $function_id = $original_function_id;
+            }
+
+            $namespaced_function_exists = $codebase_functions->functionExists(
+                $statements_analyzer,
+                strtolower($function_id)
+            );
+
+            if (!$namespaced_function_exists
+                && !$stmt->name instanceof PhpParser\Node\Name\FullyQualified
+            ) {
+                $in_call_map = CallMap::inCallMap($original_function_id);
+                $is_stubbed = $codebase_functions->hasStubbedFunction($original_function_id);
+
+                if ($is_stubbed || $in_call_map) {
+                    $function_id = $original_function_id;
+                }
+            } else {
+                $in_call_map = CallMap::inCallMap($function_id);
+                $is_stubbed = $codebase_functions->hasStubbedFunction($function_id);
+            }
+
+            if ($is_stubbed || $in_call_map || $namespaced_function_exists) {
+                $function_exists = true;
+            }
 
             $is_predefined = true;
 
@@ -212,17 +241,9 @@ class FunctionCallAnalyzer extends \Psalm\Internal\Analyzer\Statements\Expressio
 
             if (!$in_call_map) {
                 $predefined_functions = $config->getPredefinedFunctions();
-                $is_predefined = isset($predefined_functions[$function_id]);
-            }
+                $is_predefined = isset($predefined_functions[strtolower($original_function_id)])
+                    || isset($predefined_functions[strtolower($function_id)]);
 
-            if (!$in_call_map && !$stmt->name instanceof PhpParser\Node\Name\FullyQualified) {
-                $function_id = $codebase_functions->getFullyQualifiedFunctionNameFromString(
-                    $function_id,
-                    $statements_analyzer
-                );
-            }
-
-            if (!$in_call_map) {
                 if ($context->check_functions) {
                     if (self::checkFunctionExists(
                         $statements_analyzer,
@@ -233,18 +254,7 @@ class FunctionCallAnalyzer extends \Psalm\Internal\Analyzer\Statements\Expressio
                     ) {
                         return false;
                     }
-                } else {
-                    $function_id = self::getExistingFunctionId(
-                        $statements_analyzer,
-                        $function_id,
-                        $is_maybe_root_function
-                    );
                 }
-
-                $function_exists = $is_stubbed || $codebase_functions->functionExists(
-                    $statements_analyzer,
-                    strtolower($function_id)
-                );
             } else {
                 $function_exists = true;
             }
diff --git a/src/Psalm/Internal/Analyzer/Statements/Expression/CallAnalyzer.php b/src/Psalm/Internal/Analyzer/Statements/Expression/CallAnalyzer.php
index cbdfc92f4..42399957e 100644
--- a/src/Psalm/Internal/Analyzer/Statements/Expression/CallAnalyzer.php
+++ b/src/Psalm/Internal/Analyzer/Statements/Expression/CallAnalyzer.php
@@ -2099,41 +2099,6 @@ class CallAnalyzer
         return true;
     }
 
-    /**
-     * @param  StatementsAnalyzer    $statements_analyzer
-     * @param  string               $function_id
-     * @param  bool                 $can_be_in_root_scope if true, the function can be shortened to the root version
-     *
-     * @return string
-     */
-    protected static function getExistingFunctionId(
-        StatementsAnalyzer $statements_analyzer,
-        $function_id,
-        $can_be_in_root_scope
-    ) {
-        $function_id = strtolower($function_id);
-
-        $codebase = $statements_analyzer->getCodebase();
-
-        if ($codebase->functions->functionExists($statements_analyzer, $function_id)) {
-            return $function_id;
-        }
-
-        if (!$can_be_in_root_scope) {
-            return $function_id;
-        }
-
-        $root_function_id = preg_replace('/.*\\\/', '', $function_id);
-
-        if ($function_id !== $root_function_id
-            && $codebase->functions->functionExists($statements_analyzer, $root_function_id)
-        ) {
-            return $root_function_id;
-        }
-
-        return $function_id;
-    }
-
     /**
      * @param PhpParser\Node\Identifier|PhpParser\Node\Name $expr
      * @param  \Psalm\Storage\Assertion[] $assertions
@@ -2159,6 +2124,8 @@ class CallAnalyzer
         foreach ($assertions as $assertion) {
             $assertion_var_id = null;
 
+            $arg_value = null;
+
             if (is_int($assertion->var_id)) {
                 if (!isset($args[$assertion->var_id])) {
                     continue;
@@ -2229,6 +2196,23 @@ class CallAnalyzer
                 } else {
                     $type_assertions[$assertion_var_id] = $assertion->rule;
                 }
+            } elseif ($arg_value && $assertion->rule === [['!falsy']]) {
+                $assert_clauses = \Psalm\Type\Algebra::getFormula(
+                    $arg_value,
+                    $statements_analyzer->getFQCLN(),
+                    $statements_analyzer,
+                    $statements_analyzer->getCodebase()
+                );
+
+                $simplified_clauses = \Psalm\Type\Algebra::simplifyCNF(
+                    array_merge($context->clauses, $assert_clauses)
+                );
+
+                $assert_type_assertions = \Psalm\Type\Algebra::getTruthsFromFormula(
+                    $simplified_clauses
+                );
+
+                $type_assertions = array_merge($type_assertions, $assert_type_assertions);
             }
         }
 
diff --git a/tests/AssertTest.php b/tests/AssertTest.php
index 5a8baa5ad..cc01e80dc 100644
--- a/tests/AssertTest.php
+++ b/tests/AssertTest.php
@@ -30,6 +30,23 @@ class AssertTest extends TestCase
                         $a->foo();
                     }',
             ],
+            'dropInReplacementForAssert' => [
+                '<?php
+                    /**
+                     * @param mixed $_b
+                     * @psalm-assert !falsy $_b
+                     */
+                    function myAssert($_b) : void {
+                        if (!$_b) {
+                            throw new \Exception("bad");
+                        }
+                    }
+
+                    function bar(?string $s) : string {
+                        myAssert($s !== null);
+                        return $s;
+                    }'
+            ],
             'assertInstanceOfInterface' => [
                 '<?php
                     class A {
diff --git a/tests/FunctionCallTest.php b/tests/FunctionCallTest.php
index 2d349e871..a99cd057c 100644
--- a/tests/FunctionCallTest.php
+++ b/tests/FunctionCallTest.php
@@ -1268,6 +1268,22 @@ class FunctionCallTest extends TestCase
                         foo($a);
                     }',
             ],
+            'SKIPPED-getTypeHasValues' => [
+                '<?php
+                    /**
+                     * @param mixed $maybe
+                     */
+                    function matchesTypes($maybe) : void {
+                        $t = gettype($maybe);
+                        if ($t === "object") {}
+                    }'
+            ],
+            'functionResolutionInNamespace' => [
+                '<?php
+                    namespace Foo;
+                    function sort(int $_) : void {}
+                    sort(5);'
+            ],
         ];
     }