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Theorem bj-alcomexcom 32999
Description: Commutation of universal quantifiers implies commutation of existential quantifiers. Can be placed in the ax-4 1886 section, soon after 2nexaln 1906, and used to prove excom 2192. (Contributed by BJ, 29-Nov-2020.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-alcomexcom ((∀𝑥𝑦 ¬ 𝜑 → ∀𝑦𝑥 ¬ 𝜑) → (∃𝑦𝑥𝜑 → ∃𝑥𝑦𝜑))

Proof of Theorem bj-alcomexcom
StepHypRef Expression
1 2nexaln 1906 . . 3 (¬ ∃𝑥𝑦𝜑 ↔ ∀𝑥𝑦 ¬ 𝜑)
2 2nexaln 1906 . . 3 (¬ ∃𝑦𝑥𝜑 ↔ ∀𝑦𝑥 ¬ 𝜑)
31, 2imbi12i 339 . 2 ((¬ ∃𝑥𝑦𝜑 → ¬ ∃𝑦𝑥𝜑) ↔ (∀𝑥𝑦 ¬ 𝜑 → ∀𝑦𝑥 ¬ 𝜑))
4 con4 112 . 2 ((¬ ∃𝑥𝑦𝜑 → ¬ ∃𝑦𝑥𝜑) → (∃𝑦𝑥𝜑 → ∃𝑥𝑦𝜑))
53, 4sylbir 225 1 ((∀𝑥𝑦 ¬ 𝜑 → ∀𝑦𝑥 ¬ 𝜑) → (∃𝑦𝑥𝜑 → ∃𝑥𝑦𝜑))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1630  wex 1853
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886
This theorem depends on definitions:  df-bi 197  df-ex 1854
This theorem is referenced by: (None)
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