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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 )