 Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  ax-9 GIF version

Theorem ax-9 1424
 Description: Derive ax-9 1424 from ax-i9 1423, the modified version for intuitionistic logic. Although ax-9 1424 does hold intuistionistically, in intuitionistic logic it is weaker than ax-i9 1423. (Contributed by NM, 3-Feb-2015.)
Assertion
Ref Expression
ax-9 ¬ ∀𝑥 ¬ 𝑥 = 𝑦

Proof of Theorem ax-9
StepHypRef Expression
1 ax-i9 1423 . . 3 𝑥 𝑥 = 𝑦
21notnoti 574 . 2 ¬ ¬ ∃𝑥 𝑥 = 𝑦
3 alnex 1388 . 2 (∀𝑥 ¬ 𝑥 = 𝑦 ↔ ¬ ∃𝑥 𝑥 = 𝑦)
42, 3mtbir 596 1 ¬ ∀𝑥 ¬ 𝑥 = 𝑦
 Colors of variables: wff set class Syntax hints:  ¬ wn 3  ∀wal 1241   = wceq 1243  ∃wex 1381 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-5 1336  ax-gen 1338  ax-ie2 1383  ax-i9 1423 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-fal 1249 This theorem is referenced by:  equidqe  1425
 Copyright terms: Public domain W3C validator