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Theorem neeq1 2218
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 2046 . . 3 (𝐴 = 𝐵 → (𝐴 = 𝐶𝐵 = 𝐶))
21notbid 592 . 2 (𝐴 = 𝐵 → (¬ 𝐴 = 𝐶 ↔ ¬ 𝐵 = 𝐶))
3 df-ne 2206 . 2 (𝐴𝐶 ↔ ¬ 𝐴 = 𝐶)
4 df-ne 2206 . 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 1243  wne 2204
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-4 1400  ax-17 1419  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-cleq 2033  df-ne 2206
This theorem is referenced by:  neeq1i  2220  neeq1d  2223  nelrdva  2746  psseq1  3031  0inp0  3919  uzn0  8488  xrnemnf  8699  xrnepnf  8700  ngtmnft  8731  fztpval  8945
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