ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  neleq2 GIF version

Theorem neleq2 2319
Description: Equality theorem for negated membership. (Contributed by NM, 20-Nov-1994.)
Assertion
Ref Expression
neleq2 (𝐴 = 𝐵 → (𝐶𝐴𝐶𝐵))

Proof of Theorem neleq2
StepHypRef Expression
1 eleq2 2117 . . 3 (𝐴 = 𝐵 → (𝐶𝐴𝐶𝐵))
21notbid 602 . 2 (𝐴 = 𝐵 → (¬ 𝐶𝐴 ↔ ¬ 𝐶𝐵))
3 df-nel 2315 . 2 (𝐶𝐴 ↔ ¬ 𝐶𝐴)
4 df-nel 2315 . 2 (𝐶𝐵 ↔ ¬ 𝐶𝐵)
52, 3, 43bitr4g 216 1 (𝐴 = 𝐵 → (𝐶𝐴𝐶𝐵))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 102   = wceq 1259  wcel 1409  wnel 2314
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-ie1 1398  ax-ie2 1399  ax-4 1416  ax-17 1435  ax-ial 1443  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-cleq 2049  df-clel 2052  df-nel 2315
This theorem is referenced by:  neleq12d  2320
  Copyright terms: Public domain W3C validator