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Theorem neeq1 2264
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 2091 . . 3 (𝐴 = 𝐵 → (𝐴 = 𝐶𝐵 = 𝐶))
21notbid 625 . 2 (𝐴 = 𝐵 → (¬ 𝐴 = 𝐶 ↔ ¬ 𝐵 = 𝐶))
3 df-ne 2252 . 2 (𝐴𝐶 ↔ ¬ 𝐴 = 𝐶)
4 df-ne 2252 . 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 1287  wne 2251
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 1379  ax-gen 1381  ax-4 1443  ax-17 1462  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-cleq 2078  df-ne 2252
This theorem is referenced by:  neeq1i  2266  neeq1d  2269  nelrdva  2811  0inp0  3976  frecabcl  6118  eldju2ndl  6707  updjudhf  6714  xnn0nemnf  8680  uzn0  8966  xrnemnf  9180  xrnepnf  9181  ngtmnft  9212  fztpval  9427
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