ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  neeq2 GIF version

Theorem neeq2 2234
Description: Equality theorem for inequality. (Contributed by NM, 19-Nov-1994.)
Assertion
Ref Expression
neeq2 (𝐴 = 𝐵 → (𝐶𝐴𝐶𝐵))

Proof of Theorem neeq2
StepHypRef Expression
1 eqeq2 2065 . . 3 (𝐴 = 𝐵 → (𝐶 = 𝐴𝐶 = 𝐵))
21notbid 602 . 2 (𝐴 = 𝐵 → (¬ 𝐶 = 𝐴 ↔ ¬ 𝐶 = 𝐵))
3 df-ne 2221 . 2 (𝐶𝐴 ↔ ¬ 𝐶 = 𝐴)
4 df-ne 2221 . 2 (𝐶𝐵 ↔ ¬ 𝐶 = 𝐵)
52, 3, 43bitr4g 216 1 (𝐴 = 𝐵 → (𝐶𝐴𝐶𝐵))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 102   = wceq 1259  wne 2220
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-5 1352  ax-gen 1354  ax-4 1416  ax-17 1435  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-cleq 2049  df-ne 2221
This theorem is referenced by:  neeq2i  2236  neeq2d  2239  psseq2  3059  xrlttri3  8818
  Copyright terms: Public domain W3C validator