Theorem bj-alcomexcom 34789

Description: Commutation of universal quantifiers implies commutation of existential quantifiers. Can be placed in the ax-4 1813 section, soon after 2nexaln 1833, and used to prove excom 2164. (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		2nexaln 1833	. . 3 ⊢ (¬ ∃𝑥∃𝑦𝜑 ↔ ∀𝑥∀𝑦 ¬ 𝜑)
2		2nexaln 1833	. . 3 ⊢ (¬ ∃𝑦∃𝑥𝜑 ↔ ∀𝑦∀𝑥 ¬ 𝜑)
3	1, 2	imbi12i 350	. 2 ⊢ ((¬ ∃𝑥∃𝑦𝜑 → ¬ ∃𝑦∃𝑥𝜑) ↔ (∀𝑥∀𝑦 ¬ 𝜑 → ∀𝑦∀𝑥 ¬ 𝜑))
4		con4 113	. 2 ⊢ ((¬ ∃𝑥∃𝑦𝜑 → ¬ ∃𝑦∃𝑥𝜑) → (∃𝑦∃𝑥𝜑 → ∃𝑥∃𝑦𝜑))
5	3, 4	sylbir 234	1 ⊢ ((∀𝑥∀𝑦 ¬ 𝜑 → ∀𝑦∀𝑥 ¬ 𝜑) → (∃𝑦∃𝑥𝜑 → ∃𝑥∃𝑦𝜑))

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1537  ∃wex 1783

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813

This theorem depends on definitions:  df-bi 206  df-ex 1784

This theorem is referenced by: (None)