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Theorem neeq2 2296
 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 2124 . . 3 (𝐴 = 𝐵 → (𝐶 = 𝐴𝐶 = 𝐵))
21notbid 639 . 2 (𝐴 = 𝐵 → (¬ 𝐶 = 𝐴 ↔ ¬ 𝐶 = 𝐵))
3 df-ne 2283 . 2 (𝐶𝐴 ↔ ¬ 𝐶 = 𝐴)
4 df-ne 2283 . 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 1314   ≠ wne 2282 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-5 1406  ax-gen 1408  ax-4 1470  ax-17 1489  ax-ext 2097 This theorem depends on definitions:  df-bi 116  df-cleq 2108  df-ne 2283 This theorem is referenced by:  neeq2i  2298  neeq2d  2301  disji2  3888  fodjuomnilemdc  6966  xrlttri3  9476
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