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

Theorem equidqe 1425
 Description: equid 1589 with some quantification and negation without using ax-4 1400 or ax-17 1419. (Contributed by NM, 13-Jan-2011.) (Proof shortened by Wolf Lammen, 27-Feb-2014.)
Assertion
Ref Expression
equidqe ¬ ∀𝑦 ¬ 𝑥 = 𝑥

Proof of Theorem equidqe
StepHypRef Expression
1 ax-9 1424 . 2 ¬ ∀𝑦 ¬ 𝑦 = 𝑥
2 ax-8 1395 . . . . 5 (𝑦 = 𝑥 → (𝑦 = 𝑥𝑥 = 𝑥))
32pm2.43i 43 . . . 4 (𝑦 = 𝑥𝑥 = 𝑥)
43con3i 562 . . 3 𝑥 = 𝑥 → ¬ 𝑦 = 𝑥)
54alimi 1344 . 2 (∀𝑦 ¬ 𝑥 = 𝑥 → ∀𝑦 ¬ 𝑦 = 𝑥)
61, 5mto 588 1 ¬ ∀𝑦 ¬ 𝑥 = 𝑥
 Colors of variables: wff set class Syntax hints:  ¬ wn 3  ∀wal 1241 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-8 1395  ax-i9 1423 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-fal 1249 This theorem is referenced by:  ax4sp1  1426
 Copyright terms: Public domain W3C validator