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Theorem nnedc 2225
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 2221 . . . 4 (𝐴𝐵 ↔ ¬ 𝐴 = 𝐵)
21a1i 9 . . 3 (DECID 𝐴 = 𝐵 → (𝐴𝐵 ↔ ¬ 𝐴 = 𝐵))
32con2biidc 784 . 2 (DECID 𝐴 = 𝐵 → (𝐴 = 𝐵 ↔ ¬ 𝐴𝐵))
43bicomd 133 1 (DECID 𝐴 = 𝐵 → (¬ 𝐴𝐵𝐴 = 𝐵))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 102  DECID wdc 753   = wceq 1259  wne 2220
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640
This theorem depends on definitions:  df-bi 114  df-dc 754  df-ne 2221
This theorem is referenced by:  nn0n0n1ge2b  8378  alzdvds  10166  fzo0dvdseq  10169  algcvgblem  10271
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