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Theorem bj-alcomexcom 35174
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 1812 section, soon after 2nexaln 1833, and used to prove excom 2163. (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 con34b 316 . 2 ((∃𝑦𝑥𝜑 → ∃𝑥𝑦𝜑) ↔ (¬ ∃𝑥𝑦𝜑 → ¬ ∃𝑦𝑥𝜑))
2 2nexaln 1833 . . 3 (¬ ∃𝑥𝑦𝜑 ↔ ∀𝑥𝑦 ¬ 𝜑)
3 2nexaln 1833 . . 3 (¬ ∃𝑦𝑥𝜑 ↔ ∀𝑦𝑥 ¬ 𝜑)
42, 3imbi12i 351 . 2 ((¬ ∃𝑥𝑦𝜑 → ¬ ∃𝑦𝑥𝜑) ↔ (∀𝑥𝑦 ¬ 𝜑 → ∀𝑦𝑥 ¬ 𝜑))
51, 4bitr2i 276 1 ((∀𝑥𝑦 ¬ 𝜑 → ∀𝑦𝑥 ¬ 𝜑) ↔ (∃𝑦𝑥𝜑 → ∃𝑥𝑦𝜑))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wal 1540  wex 1782
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812
This theorem depends on definitions:  df-bi 206  df-ex 1783
This theorem is referenced by: (None)
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