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Theorem neeq2 2219
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 2049 . . 3 (𝐴 = 𝐵 → (𝐶 = 𝐴𝐶 = 𝐵))
21notbid 592 . 2 (𝐴 = 𝐵 → (¬ 𝐶 = 𝐴 ↔ ¬ 𝐶 = 𝐵))
3 df-ne 2206 . 2 (𝐶𝐴 ↔ ¬ 𝐶 = 𝐴)
4 df-ne 2206 . 2 (𝐶𝐵 ↔ ¬ 𝐶 = 𝐵)
52, 3, 43bitr4g 212 1 (𝐴 = 𝐵 → (𝐶𝐴𝐶𝐵))
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:  neeq2i  2221  neeq2d  2224  psseq2  3032  xrlttri3  8718
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