Theorem bj-alcomexcom 36622

Description: Commutation of two existential quantifiers on a formula is equivalent to commutation of two universal quantifiers over the same variables on the negation of that formula. Can be placed in the ax-4 1808 section, soon after 2nexaln 1829, and used to prove excom 2161. (Contributed by BJ, 29-Nov-2020.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-alcomexcom	⊢ ((∀𝑥∀𝑦 ¬ 𝜑 → ∀𝑦∀𝑥 ¬ 𝜑) ↔ (∃𝑦∃𝑥𝜑 → ∃𝑥∃𝑦𝜑))

Proof of Theorem bj-alcomexcom

Step	Hyp	Ref	Expression
1		con34b 316	. 2 ⊢ ((∃𝑦∃𝑥𝜑 → ∃𝑥∃𝑦𝜑) ↔ (¬ ∃𝑥∃𝑦𝜑 → ¬ ∃𝑦∃𝑥𝜑))
2		2nexaln 1829	. . 3 ⊢ (¬ ∃𝑥∃𝑦𝜑 ↔ ∀𝑥∀𝑦 ¬ 𝜑)
3		2nexaln 1829	. . 3 ⊢ (¬ ∃𝑦∃𝑥𝜑 ↔ ∀𝑦∀𝑥 ¬ 𝜑)
4	2, 3	imbi12i 350	. 2 ⊢ ((¬ ∃𝑥∃𝑦𝜑 → ¬ ∃𝑦∃𝑥𝜑) ↔ (∀𝑥∀𝑦 ¬ 𝜑 → ∀𝑦∀𝑥 ¬ 𝜑))
5	1, 4	bitr2i 276	1 ⊢ ((∀𝑥∀𝑦 ¬ 𝜑 → ∀𝑦∀𝑥 ¬ 𝜑) ↔ (∃𝑦∃𝑥𝜑 → ∃𝑥∃𝑦𝜑))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206  ∀wal 1537  ∃wex 1778

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808

This theorem depends on definitions:  df-bi 207  df-ex 1779

This theorem is referenced by: (None)