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Theorem nnedc 2332
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 2328 . . . 4 (𝐴𝐵 ↔ ¬ 𝐴 = 𝐵)
21a1i 9 . . 3 (DECID 𝐴 = 𝐵 → (𝐴𝐵 ↔ ¬ 𝐴 = 𝐵))
32con2biidc 865 . 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 820   = wceq 1335  wne 2327
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 604  ax-in2 605  ax-io 699
This theorem depends on definitions:  df-bi 116  df-dc 821  df-ne 2328
This theorem is referenced by:  nn0n0n1ge2b  9248  alzdvds  11758  fzo0dvdseq  11761  algcvgblem  11941
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