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Theorem axc5c711 33683
Description: Proof of a single axiom that can replace ax-c5 33648, ax-c7 33650, and ax-11 2031 in a subsystem that includes these axioms plus ax-c4 33649 and ax-gen 1719 (and propositional calculus). See axc5c711toc5 33684, axc5c711toc7 33685, and axc5c711to11 33686 for the rederivation of those axioms. This theorem extends the idea in Scott Fenton's axc5c7 33676. (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 33648 . . 3 (∀𝑦𝜑𝜑)
2 ax10fromc7 33660 . . . 4 (¬ ∀𝑦𝜑 → ∀𝑦 ¬ ∀𝑦𝜑)
3 ax-c7 33650 . . . . . 6 (¬ ∀𝑥 ¬ ∀𝑥𝑦𝜑 → ∀𝑦𝜑)
43con1i 144 . . . . 5 (¬ ∀𝑦𝜑 → ∀𝑥 ¬ ∀𝑥𝑦𝜑)
54alimi 1736 . . . 4 (∀𝑦 ¬ ∀𝑦𝜑 → ∀𝑦𝑥 ¬ ∀𝑥𝑦𝜑)
6 ax-11 2031 . . . 4 (∀𝑦𝑥 ¬ ∀𝑥𝑦𝜑 → ∀𝑥𝑦 ¬ ∀𝑥𝑦𝜑)
72, 5, 63syl 18 . . 3 (¬ ∀𝑦𝜑 → ∀𝑥𝑦 ¬ ∀𝑥𝑦𝜑)
81, 7nsyl4 156 . 2 (¬ ∀𝑥𝑦 ¬ ∀𝑥𝑦𝜑𝜑)
9 ax-c5 33648 . 2 (∀𝑥𝜑𝜑)
108, 9ja 173 1 ((∀𝑥𝑦 ¬ ∀𝑥𝑦𝜑 → ∀𝑥𝜑) → 𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wal 1478
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-11 2031  ax-c5 33648  ax-c4 33649  ax-c7 33650
This theorem is referenced by:  axc5c711toc5  33684  axc5c711toc7  33685  axc5c711to11  33686
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