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Theorem nnedc 2254
 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 2250 . . . 4 (𝐴𝐵 ↔ ¬ 𝐴 = 𝐵)
21a1i 9 . . 3 (DECID 𝐴 = 𝐵 → (𝐴𝐵 ↔ ¬ 𝐴 = 𝐵))
32con2biidc 807 . 2 (DECID 𝐴 = 𝐵 → (𝐴 = 𝐵 ↔ ¬ 𝐴𝐵))
43bicomd 139 1 (DECID 𝐴 = 𝐵 → (¬ 𝐴𝐵𝐴 = 𝐵))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 103  DECID wdc 776   = wceq 1285   ≠ wne 2249 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663 This theorem depends on definitions:  df-bi 115  df-dc 777  df-ne 2250 This theorem is referenced by:  nn0n0n1ge2b  8722  alzdvds  10635  fzo0dvdseq  10638  algcvgblem  10811
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