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Theorem axc711 34518
 Description: Proof of a single axiom that can replace both ax-c7 34489 and ax-11 2074. See axc711toc7 34520 and axc711to11 34521 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 2074 . . . . 5 (∀𝑦𝑥𝜑 → ∀𝑥𝑦𝜑)
21con3i 150 . . . 4 (¬ ∀𝑥𝑦𝜑 → ¬ ∀𝑦𝑥𝜑)
32alimi 1779 . . 3 (∀𝑥 ¬ ∀𝑥𝑦𝜑 → ∀𝑥 ¬ ∀𝑦𝑥𝜑)
43con3i 150 . 2 (¬ ∀𝑥 ¬ ∀𝑦𝑥𝜑 → ¬ ∀𝑥 ¬ ∀𝑥𝑦𝜑)
5 ax-c7 34489 . 2 (¬ ∀𝑥 ¬ ∀𝑥𝑦𝜑 → ∀𝑦𝜑)
64, 5syl 17 1 (¬ ∀𝑥 ¬ ∀𝑦𝑥𝜑 → ∀𝑦𝜑)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1521 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-11 2074  ax-c7 34489 This theorem is referenced by:  axc711toc7  34520  axc711to11  34521
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