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Theorem neleq1 2301
 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 2100 . . 3 (𝐴 = 𝐵 → (𝐴𝐶𝐵𝐶))
21notbid 592 . 2 (𝐴 = 𝐵 → (¬ 𝐴𝐶 ↔ ¬ 𝐵𝐶))
3 df-nel 2207 . 2 (𝐴𝐶 ↔ ¬ 𝐴𝐶)
4 df-nel 2207 . 2 (𝐵𝐶 ↔ ¬ 𝐵𝐶)
52, 3, 43bitr4g 212 1 (𝐴 = 𝐵 → (𝐴𝐶𝐵𝐶))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 98   = wceq 1243   ∈ wcel 1393   ∉ wnel 2205 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-17 1419  ax-ial 1427  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-cleq 2033  df-clel 2036  df-nel 2207 This theorem is referenced by:  neleq12d  2303  ruALT  4275
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