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Theorem neleq1 2407
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 2202 . . 3 (𝐴 = 𝐵 → (𝐴𝐶𝐵𝐶))
21notbid 656 . 2 (𝐴 = 𝐵 → (¬ 𝐴𝐶 ↔ ¬ 𝐵𝐶))
3 df-nel 2404 . 2 (𝐴𝐶 ↔ ¬ 𝐴𝐶)
4 df-nel 2404 . 2 (𝐵𝐶 ↔ ¬ 𝐵𝐶)
52, 3, 43bitr4g 222 1 (𝐴 = 𝐵 → (𝐴𝐶𝐵𝐶))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 104   = wceq 1331  wcel 1480  wnel 2403
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-5 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-4 1487  ax-17 1506  ax-ial 1514  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-cleq 2132  df-clel 2135  df-nel 2404
This theorem is referenced by:  neleq12d  2409  ruALT  4466
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