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

Theorem equidqe 1468
Description: equid 1632 with some quantification and negation without using ax-4 1443 or ax-17 1462. (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 1467 . 2 ¬ ∀𝑦 ¬ 𝑦 = 𝑥
2 ax-8 1438 . . . . 5 (𝑦 = 𝑥 → (𝑦 = 𝑥𝑥 = 𝑥))
32pm2.43i 48 . . . 4 (𝑦 = 𝑥𝑥 = 𝑥)
43con3i 595 . . 3 𝑥 = 𝑥 → ¬ 𝑦 = 𝑥)
54alimi 1387 . 2 (∀𝑦 ¬ 𝑥 = 𝑥 → ∀𝑦 ¬ 𝑦 = 𝑥)
61, 5mto 621 1 ¬ ∀𝑦 ¬ 𝑥 = 𝑥
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wal 1285
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-5 1379  ax-gen 1381  ax-ie2 1426  ax-8 1438  ax-i9 1466
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-fal 1293
This theorem is referenced by:  ax4sp1  1469
  Copyright terms: Public domain W3C validator