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Theorem neleq2 2409
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 2204 . . 3 (𝐴 = 𝐵 → (𝐶𝐴𝐶𝐵))
21notbid 657 . 2 (𝐴 = 𝐵 → (¬ 𝐶𝐴 ↔ ¬ 𝐶𝐵))
3 df-nel 2405 . 2 (𝐶𝐴 ↔ ¬ 𝐶𝐴)
4 df-nel 2405 . 2 (𝐶𝐵 ↔ ¬ 𝐶𝐵)
52, 3, 43bitr4g 222 1 (𝐴 = 𝐵 → (𝐶𝐴𝐶𝐵))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 104   = wceq 1332  wcel 1481  wnel 2404
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 604  ax-in2 605  ax-5 1424  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-4 1488  ax-17 1507  ax-ial 1515  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-cleq 2133  df-clel 2136  df-nel 2405
This theorem is referenced by:  neleq12d  2410
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