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Theorem dcned 2291
Description: Decidable equality implies decidable negated equality. (Contributed by Jim Kingdon, 3-May-2020.)
Hypothesis
Ref Expression
dcned.eq (𝜑DECID 𝐴 = 𝐵)
Assertion
Ref Expression
dcned (𝜑DECID 𝐴𝐵)

Proof of Theorem dcned
StepHypRef Expression
1 dcned.eq . . 3 (𝜑DECID 𝐴 = 𝐵)
2 dcn 812 . . 3 (DECID 𝐴 = 𝐵DECID ¬ 𝐴 = 𝐵)
31, 2syl 14 . 2 (𝜑DECID ¬ 𝐴 = 𝐵)
4 df-ne 2286 . . 3 (𝐴𝐵 ↔ ¬ 𝐴 = 𝐵)
54dcbii 810 . 2 (DECID 𝐴𝐵DECID ¬ 𝐴 = 𝐵)
63, 5sylibr 133 1 (𝜑DECID 𝐴𝐵)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  DECID wdc 804   = wceq 1316  wne 2285
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 588  ax-in2 589  ax-io 683
This theorem depends on definitions:  df-bi 116  df-dc 805  df-ne 2286
This theorem is referenced by:  nn0n0n1ge2b  9098  algcvgblem  11657
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