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Database ZF (ZERMELO-FRAENKEL) SET THEORY ZF Set Theory - start with the Axiom of Extensionality Restricted quantification Restricted universal and existential quantification rexcom
Description: Commutation of restricted existential quantifiers. (Contributed by NM , 19-Nov-1995) (Revised by Mario Carneiro , 14-Oct-2016) (Proof
shortened by BJ , 26-Aug-2023) (Proof shortened by Wolf Lammen , 8-Dec-2024)
Ref
Expression
Assertion
rexcom ⊢ ∃ x ∈ A ∃ y ∈ B φ ↔ ∃ y ∈ B ∃ x ∈ A φ
Proof
Step
Hyp
Ref
Expression
1
ralcom ⊢ ∀ x ∈ A ∀ y ∈ B ¬ φ ↔ ∀ y ∈ B ∀ x ∈ A ¬ φ
2
ralnex2 ⊢ ∀ x ∈ A ∀ y ∈ B ¬ φ ↔ ¬ ∃ x ∈ A ∃ y ∈ B φ
3
ralnex2 ⊢ ∀ y ∈ B ∀ x ∈ A ¬ φ ↔ ¬ ∃ y ∈ B ∃ x ∈ A φ
4
1 2 3
3bitr3i ⊢ ¬ ∃ x ∈ A ∃ y ∈ B φ ↔ ¬ ∃ y ∈ B ∃ x ∈ A φ
5
4
con4bii ⊢ ∃ x ∈ A ∃ y ∈ B φ ↔ ∃ y ∈ B ∃ x ∈ A φ