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Theorem dcned 2226
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 757 . . 3 (DECID 𝐴 = 𝐵DECID ¬ 𝐴 = 𝐵)
31, 2syl 14 . 2 (𝜑DECID ¬ 𝐴 = 𝐵)
4 df-ne 2221 . . 3 (𝐴𝐵 ↔ ¬ 𝐴 = 𝐵)
54dcbii 758 . 2 (DECID 𝐴𝐵DECID ¬ 𝐴 = 𝐵)
63, 5sylibr 141 1 (𝜑DECID 𝐴𝐵)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  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  algcvgblem  10271
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