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Theorem nnedc 2211
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 2206 . . . 4 (𝐴𝐵 ↔ ¬ 𝐴 = 𝐵)
21a1i 9 . . 3 (DECID 𝐴 = 𝐵 → (𝐴𝐵 ↔ ¬ 𝐴 = 𝐵))
32con2biidc 773 . 2 (DECID 𝐴 = 𝐵 → (𝐴 = 𝐵 ↔ ¬ 𝐴𝐵))
43bicomd 129 1 (DECID 𝐴 = 𝐵 → (¬ 𝐴𝐵𝐴 = 𝐵))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 98  DECID wdc 742   = wceq 1243  wne 2204
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630
This theorem depends on definitions:  df-bi 110  df-dc 743  df-ne 2206
This theorem is referenced by:  nn0n0n1ge2b  8318  algcvgblem  9861
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