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

Theorem ax-9 1470
 Description: Derive ax-9 1470 from ax-i9 1469, the modified version for intuitionistic logic. Although ax-9 1470 does hold intuistionistically, in intuitionistic logic it is weaker than ax-i9 1469. (Contributed by NM, 3-Feb-2015.)
Assertion
Ref Expression
ax-9 ¬ ∀𝑥 ¬ 𝑥 = 𝑦

Proof of Theorem ax-9
StepHypRef Expression
1 ax-i9 1469 . . 3 𝑥 𝑥 = 𝑦
21notnoti 610 . 2 ¬ ¬ ∃𝑥 𝑥 = 𝑦
3 alnex 1434 . 2 (∀𝑥 ¬ 𝑥 = 𝑦 ↔ ¬ ∃𝑥 𝑥 = 𝑦)
42, 3mtbir 632 1 ¬ ∀𝑥 ¬ 𝑥 = 𝑦
 Colors of variables: wff set class Syntax hints:  ¬ wn 3  ∀wal 1288   = wceq 1290  ∃wex 1427 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-5 1382  ax-gen 1384  ax-ie2 1429  ax-i9 1469 This theorem depends on definitions:  df-bi 116  df-tru 1293  df-fal 1296 This theorem is referenced by:  equidqe  1471
 Copyright terms: Public domain W3C validator