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Theorem neleq1 2354
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 2150 . . 3 (𝐴 = 𝐵 → (𝐴𝐶𝐵𝐶))
21notbid 627 . 2 (𝐴 = 𝐵 → (¬ 𝐴𝐶 ↔ ¬ 𝐵𝐶))
3 df-nel 2351 . 2 (𝐴𝐶 ↔ ¬ 𝐴𝐶)
4 df-nel 2351 . 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 1289  wcel 1438  wnel 2350
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 579  ax-in2 580  ax-5 1381  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-4 1445  ax-17 1464  ax-ial 1472  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-cleq 2081  df-clel 2084  df-nel 2351
This theorem is referenced by:  neleq12d  2356  ruALT  4357
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