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Theorem neeq1 2233
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 2062 . . 3 (𝐴 = 𝐵 → (𝐴 = 𝐶𝐵 = 𝐶))
21notbid 602 . 2 (𝐴 = 𝐵 → (¬ 𝐴 = 𝐶 ↔ ¬ 𝐵 = 𝐶))
3 df-ne 2221 . 2 (𝐴𝐶 ↔ ¬ 𝐴 = 𝐶)
4 df-ne 2221 . 2 (𝐵𝐶 ↔ ¬ 𝐵 = 𝐶)
52, 3, 43bitr4g 216 1 (𝐴 = 𝐵 → (𝐴𝐶𝐵𝐶))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 102   = wceq 1259  wne 2220
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-5 1352  ax-gen 1354  ax-4 1416  ax-17 1435  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-cleq 2049  df-ne 2221
This theorem is referenced by:  neeq1i  2235  neeq1d  2238  nelrdva  2768  psseq1  3058  0inp0  3946  uzn0  8583  xrnemnf  8799  xrnepnf  8800  ngtmnft  8831  fztpval  9046
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