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Theorem neeq1 2213
 Description: Equality theorem for inequality. (Contributed by NM, 19-Nov-1994.)
Assertion
Ref Expression
neeq1 (A = B → (A𝐶B𝐶))

Proof of Theorem neeq1
StepHypRef Expression
1 eqeq1 2043 . . 3 (A = B → (A = 𝐶B = 𝐶))
21notbid 591 . 2 (A = B → (¬ A = 𝐶 ↔ ¬ B = 𝐶))
3 df-ne 2203 . 2 (A𝐶 ↔ ¬ A = 𝐶)
4 df-ne 2203 . 2 (B𝐶 ↔ ¬ B = 𝐶)
52, 3, 43bitr4g 212 1 (A = B → (A𝐶B𝐶))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 98   = wceq 1242   ≠ wne 2201 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 1333  ax-gen 1335  ax-4 1397  ax-17 1416  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-cleq 2030  df-ne 2203 This theorem is referenced by:  neeq1i  2215  neeq1d  2218  nelrdva  2740  psseq1  3025  0inp0  3910  uzn0  8244  xrnemnf  8449  xrnepnf  8450  ngtmnft  8481  fztpval  8695
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