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Theorem axc5c711 34726
Description: Proof of a single axiom that can replace ax-c5 34691, ax-c7 34693, and ax-11 2190 in a subsystem that includes these axioms plus ax-c4 34692 and ax-gen 1870 (and propositional calculus). See axc5c711toc5 34727, axc5c711toc7 34728, and axc5c711to11 34729 for the rederivation of those axioms. This theorem extends the idea in Scott Fenton's axc5c7 34719. (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 34691 . . 3 (∀𝑦𝜑𝜑)
2 ax10fromc7 34703 . . . 4 (¬ ∀𝑦𝜑 → ∀𝑦 ¬ ∀𝑦𝜑)
3 ax-c7 34693 . . . . . 6 (¬ ∀𝑥 ¬ ∀𝑥𝑦𝜑 → ∀𝑦𝜑)
43con1i 146 . . . . 5 (¬ ∀𝑦𝜑 → ∀𝑥 ¬ ∀𝑥𝑦𝜑)
54alimi 1887 . . . 4 (∀𝑦 ¬ ∀𝑦𝜑 → ∀𝑦𝑥 ¬ ∀𝑥𝑦𝜑)
6 ax-11 2190 . . . 4 (∀𝑦𝑥 ¬ ∀𝑥𝑦𝜑 → ∀𝑥𝑦 ¬ ∀𝑥𝑦𝜑)
72, 5, 63syl 18 . . 3 (¬ ∀𝑦𝜑 → ∀𝑥𝑦 ¬ ∀𝑥𝑦𝜑)
81, 7nsyl4 157 . 2 (¬ ∀𝑥𝑦 ¬ ∀𝑥𝑦𝜑𝜑)
9 ax-c5 34691 . 2 (∀𝑥𝜑𝜑)
108, 9ja 174 1 ((∀𝑥𝑦 ¬ ∀𝑥𝑦𝜑 → ∀𝑥𝜑) → 𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wal 1629
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-11 2190  ax-c5 34691  ax-c4 34692  ax-c7 34693
This theorem is referenced by:  axc5c711toc5  34727  axc5c711toc7  34728  axc5c711to11  34729
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