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Theorem axc711 33217
Description: Proof of a single axiom that can replace both ax-c7 33188 and ax-11 2021. See axc711toc7 33219 and axc711to11 33220 for the rederivation of those axioms. (Contributed by NM, 18-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
axc711 (¬ ∀𝑥 ¬ ∀𝑦𝑥𝜑 → ∀𝑦𝜑)

Proof of Theorem axc711
StepHypRef Expression
1 ax-11 2021 . . . . 5 (∀𝑦𝑥𝜑 → ∀𝑥𝑦𝜑)
21con3i 149 . . . 4 (¬ ∀𝑥𝑦𝜑 → ¬ ∀𝑦𝑥𝜑)
32alimi 1730 . . 3 (∀𝑥 ¬ ∀𝑥𝑦𝜑 → ∀𝑥 ¬ ∀𝑦𝑥𝜑)
43con3i 149 . 2 (¬ ∀𝑥 ¬ ∀𝑦𝑥𝜑 → ¬ ∀𝑥 ¬ ∀𝑥𝑦𝜑)
5 ax-c7 33188 . 2 (¬ ∀𝑥 ¬ ∀𝑥𝑦𝜑 → ∀𝑦𝜑)
64, 5syl 17 1 (¬ ∀𝑥 ¬ ∀𝑦𝑥𝜑 → ∀𝑦𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1473
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-11 2021  ax-c7 33188
This theorem is referenced by:  axc711toc7  33219  axc711to11  33220
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