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Theorem neleq2 2280
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 2083 . . 3 (A = B → (𝐶 A𝐶 B))
21notbid 579 . 2 (A = B → (¬ 𝐶 A ↔ ¬ 𝐶 B))
3 df-nel 2189 . 2 (𝐶A ↔ ¬ 𝐶 A)
4 df-nel 2189 . 2 (𝐶B ↔ ¬ 𝐶 B)
52, 3, 43bitr4g 212 1 (A = B → (𝐶A𝐶B))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 98   = wceq 1228   wcel 1374  wnel 2187
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 532  ax-in2 533  ax-5 1316  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-4 1381  ax-17 1400  ax-ial 1409  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-cleq 2015  df-clel 2018  df-nel 2189
This theorem is referenced by:  neleq12d  2281
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