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Theorem neeq2 2299
Description: Equality theorem for inequality. (Contributed by NM, 19-Nov-1994.)
Assertion
Ref Expression
neeq2 (𝐴 = 𝐵 → (𝐶𝐴𝐶𝐵))

Proof of Theorem neeq2
StepHypRef Expression
1 eqeq2 2127 . . 3 (𝐴 = 𝐵 → (𝐶 = 𝐴𝐶 = 𝐵))
21notbid 641 . 2 (𝐴 = 𝐵 → (¬ 𝐶 = 𝐴 ↔ ¬ 𝐶 = 𝐵))
3 df-ne 2286 . 2 (𝐶𝐴 ↔ ¬ 𝐶 = 𝐴)
4 df-ne 2286 . 2 (𝐶𝐵 ↔ ¬ 𝐶 = 𝐵)
52, 3, 43bitr4g 222 1 (𝐴 = 𝐵 → (𝐶𝐴𝐶𝐵))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 104   = wceq 1316  wne 2285
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-5 1408  ax-gen 1410  ax-4 1472  ax-17 1491  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-cleq 2110  df-ne 2286
This theorem is referenced by:  neeq2i  2301  neeq2d  2304  disji2  3892  fodjuomnilemdc  6984  xrlttri3  9551
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