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Theorem nnedc 2272
 Description: Negation of inequality where equality is decidable. (Contributed by Jim Kingdon, 15-May-2018.)
Assertion
Ref Expression
nnedc (DECID 𝐴 = 𝐵 → (¬ 𝐴𝐵𝐴 = 𝐵))

Proof of Theorem nnedc
StepHypRef Expression
1 df-ne 2268 . . . 4 (𝐴𝐵 ↔ ¬ 𝐴 = 𝐵)
21a1i 9 . . 3 (DECID 𝐴 = 𝐵 → (𝐴𝐵 ↔ ¬ 𝐴 = 𝐵))
32con2biidc 817 . 2 (DECID 𝐴 = 𝐵 → (𝐴 = 𝐵 ↔ ¬ 𝐴𝐵))
43bicomd 140 1 (DECID 𝐴 = 𝐵 → (¬ 𝐴𝐵𝐴 = 𝐵))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 104  DECID wdc 786   = wceq 1299   ≠ wne 2267 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 584  ax-in2 585  ax-io 671 This theorem depends on definitions:  df-bi 116  df-dc 787  df-ne 2268 This theorem is referenced by:  nn0n0n1ge2b  8982  alzdvds  11347  fzo0dvdseq  11350  algcvgblem  11523
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