Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  axc711 Structured version   Visualization version   GIF version

Theorem axc711 36207
Description: Proof of a single axiom that can replace both ax-c7 36178 and ax-11 2158. See axc711toc7 36209 and axc711to11 36210 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 2158 . . . . 5 (∀𝑦𝑥𝜑 → ∀𝑥𝑦𝜑)
21con3i 157 . . . 4 (¬ ∀𝑥𝑦𝜑 → ¬ ∀𝑦𝑥𝜑)
32alimi 1813 . . 3 (∀𝑥 ¬ ∀𝑥𝑦𝜑 → ∀𝑥 ¬ ∀𝑦𝑥𝜑)
43con3i 157 . 2 (¬ ∀𝑥 ¬ ∀𝑦𝑥𝜑 → ¬ ∀𝑥 ¬ ∀𝑥𝑦𝜑)
5 ax-c7 36178 . 2 (¬ ∀𝑥 ¬ ∀𝑥𝑦𝜑 → ∀𝑦𝜑)
64, 5syl 17 1 (¬ ∀𝑥 ¬ ∀𝑦𝑥𝜑 → ∀𝑦𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1536
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-11 2158  ax-c7 36178
This theorem is referenced by:  axc711toc7  36209  axc711to11  36210
  Copyright terms: Public domain W3C validator