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Theorem neeq1 2262
 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 2089 . . 3 (𝐴 = 𝐵 → (𝐴 = 𝐶𝐵 = 𝐶))
21notbid 625 . 2 (𝐴 = 𝐵 → (¬ 𝐴 = 𝐶 ↔ ¬ 𝐵 = 𝐶))
3 df-ne 2250 . 2 (𝐴𝐶 ↔ ¬ 𝐴 = 𝐶)
4 df-ne 2250 . 2 (𝐵𝐶 ↔ ¬ 𝐵 = 𝐶)
52, 3, 43bitr4g 221 1 (𝐴 = 𝐵 → (𝐴𝐶𝐵𝐶))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 103   = wceq 1285   ≠ wne 2249 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-5 1377  ax-gen 1379  ax-4 1441  ax-17 1460  ax-ext 2065 This theorem depends on definitions:  df-bi 115  df-cleq 2076  df-ne 2250 This theorem is referenced by:  neeq1i  2264  neeq1d  2267  nelrdva  2806  0inp0  3960  frecabcl  6068  xnn0nemnf  8481  uzn0  8767  xrnemnf  8981  xrnepnf  8982  ngtmnft  9013  fztpval  9228
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