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Theorem axc5c711 34022
Description: Proof of a single axiom that can replace ax-c5 33987, ax-c7 33989, and ax-11 2032 in a subsystem that includes these axioms plus ax-c4 33988 and ax-gen 1720 (and propositional calculus). See axc5c711toc5 34023, axc5c711toc7 34024, and axc5c711to11 34025 for the rederivation of those axioms. This theorem extends the idea in Scott Fenton's axc5c7 34015. (Contributed by NM, 18-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
axc5c711 ((∀𝑥𝑦 ¬ ∀𝑥𝑦𝜑 → ∀𝑥𝜑) → 𝜑)

Proof of Theorem axc5c711
StepHypRef Expression
1 ax-c5 33987 . . 3 (∀𝑦𝜑𝜑)
2 ax10fromc7 33999 . . . 4 (¬ ∀𝑦𝜑 → ∀𝑦 ¬ ∀𝑦𝜑)
3 ax-c7 33989 . . . . . 6 (¬ ∀𝑥 ¬ ∀𝑥𝑦𝜑 → ∀𝑦𝜑)
43con1i 144 . . . . 5 (¬ ∀𝑦𝜑 → ∀𝑥 ¬ ∀𝑥𝑦𝜑)
54alimi 1737 . . . 4 (∀𝑦 ¬ ∀𝑦𝜑 → ∀𝑦𝑥 ¬ ∀𝑥𝑦𝜑)
6 ax-11 2032 . . . 4 (∀𝑦𝑥 ¬ ∀𝑥𝑦𝜑 → ∀𝑥𝑦 ¬ ∀𝑥𝑦𝜑)
72, 5, 63syl 18 . . 3 (¬ ∀𝑦𝜑 → ∀𝑥𝑦 ¬ ∀𝑥𝑦𝜑)
81, 7nsyl4 156 . 2 (¬ ∀𝑥𝑦 ¬ ∀𝑥𝑦𝜑𝜑)
9 ax-c5 33987 . 2 (∀𝑥𝜑𝜑)
108, 9ja 173 1 ((∀𝑥𝑦 ¬ ∀𝑥𝑦𝜑 → ∀𝑥𝜑) → 𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wal 1479
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-11 2032  ax-c5 33987  ax-c4 33988  ax-c7 33989
This theorem is referenced by:  axc5c711toc5  34023  axc5c711toc7  34024  axc5c711to11  34025
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