Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  axc5c711 Structured version   Visualization version   GIF version

Theorem axc5c711 32722
Description: Proof of a single axiom that can replace ax-c5 32688, ax-c7 32690, and ax-11 1970 in a subsystem that includes these axioms plus ax-c4 32689 and ax-gen 1698 (and propositional calculus). See axc5c711toc5 32723, axc5c711toc7 32724, and axc5c711to11 32725 for the rederivation of those axioms. This theorem extends the idea in Scott Fenton's axc5c7 32715. (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 32688 . . 3 (∀𝑦𝜑𝜑)
2 ax10 32700 . . . 4 (¬ ∀𝑦𝜑 → ∀𝑦 ¬ ∀𝑦𝜑)
3 ax-c7 32690 . . . . . 6 (¬ ∀𝑥 ¬ ∀𝑥𝑦𝜑 → ∀𝑦𝜑)
43con1i 136 . . . . 5 (¬ ∀𝑦𝜑 → ∀𝑥 ¬ ∀𝑥𝑦𝜑)
54alimi 1713 . . . 4 (∀𝑦 ¬ ∀𝑦𝜑 → ∀𝑦𝑥 ¬ ∀𝑥𝑦𝜑)
6 ax-11 1970 . . . 4 (∀𝑦𝑥 ¬ ∀𝑥𝑦𝜑 → ∀𝑥𝑦 ¬ ∀𝑥𝑦𝜑)
72, 5, 63syl 18 . . 3 (¬ ∀𝑦𝜑 → ∀𝑥𝑦 ¬ ∀𝑥𝑦𝜑)
81, 7nsyl4 151 . 2 (¬ ∀𝑥𝑦 ¬ ∀𝑥𝑦𝜑𝜑)
9 ax-c5 32688 . 2 (∀𝑥𝜑𝜑)
108, 9ja 168 1 ((∀𝑥𝑦 ¬ ∀𝑥𝑦𝜑 → ∀𝑥𝜑) → 𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wal 1466
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1698  ax-4 1711  ax-11 1970  ax-c5 32688  ax-c4 32689  ax-c7 32690
This theorem is referenced by:  axc5c711toc5  32723  axc5c711toc7  32724  axc5c711to11  32725
  Copyright terms: Public domain W3C validator