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Theorem neleq1 2459
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 2252 . . 3 (𝐴 = 𝐵 → (𝐴𝐶𝐵𝐶))
21notbid 668 . 2 (𝐴 = 𝐵 → (¬ 𝐴𝐶 ↔ ¬ 𝐵𝐶))
3 df-nel 2456 . 2 (𝐴𝐶 ↔ ¬ 𝐴𝐶)
4 df-nel 2456 . 2 (𝐵𝐶 ↔ ¬ 𝐵𝐶)
52, 3, 43bitr4g 223 1 (𝐴 = 𝐵 → (𝐴𝐶𝐵𝐶))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 105   = wceq 1364  wcel 2160  wnel 2455
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-5 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-4 1521  ax-17 1537  ax-ial 1545  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-cleq 2182  df-clel 2185  df-nel 2456
This theorem is referenced by:  neleq12d  2461  ruALT  4568
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