ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  neleq1 GIF version

Theorem neleq1 2344
Description: Equality theorem for negated membership. (Contributed by NM, 20-Nov-1994.)
Assertion
Ref Expression
neleq1 (𝐴 = 𝐵 → (𝐴𝐶𝐵𝐶))

Proof of Theorem neleq1
StepHypRef Expression
1 eleq1 2142 . . 3 (𝐴 = 𝐵 → (𝐴𝐶𝐵𝐶))
21notbid 625 . 2 (𝐴 = 𝐵 → (¬ 𝐴𝐶 ↔ ¬ 𝐵𝐶))
3 df-nel 2341 . 2 (𝐴𝐶 ↔ ¬ 𝐴𝐶)
4 df-nel 2341 . 2 (𝐵𝐶 ↔ ¬ 𝐵𝐶)
52, 3, 43bitr4g 221 1 (𝐴 = 𝐵 → (𝐴𝐶𝐵𝐶))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 103   = wceq 1285  wcel 1434  wnel 2340
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 1377  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-4 1441  ax-17 1460  ax-ial 1468  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-cleq 2075  df-clel 2078  df-nel 2341
This theorem is referenced by:  neleq12d  2346  ruALT  4302
  Copyright terms: Public domain W3C validator