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Theorem bj-dcbi 12123
 Description: Equivalence property for DECID. TODO: solve conflict with dcbi 883; minimize dcbii 786 and dcbid 787 with it, as well as theorems using those. (Contributed by BJ, 27-Jan-2020.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-dcbi ((𝜑𝜓) → (DECID 𝜑DECID 𝜓))

Proof of Theorem bj-dcbi
StepHypRef Expression
1 id 19 . . 3 ((𝜑𝜓) → (𝜑𝜓))
2 bj-notbi 12120 . . 3 ((𝜑𝜓) → (¬ 𝜑 ↔ ¬ 𝜓))
31, 2orbi12d 743 . 2 ((𝜑𝜓) → ((𝜑 ∨ ¬ 𝜑) ↔ (𝜓 ∨ ¬ 𝜓)))
4 df-dc 782 . 2 (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑))
5 df-dc 782 . 2 (DECID 𝜓 ↔ (𝜓 ∨ ¬ 𝜓))
63, 4, 53bitr4g 222 1 ((𝜑𝜓) → (DECID 𝜑DECID 𝜓))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 104   ∨ wo 665  DECID wdc 781 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666 This theorem depends on definitions:  df-bi 116  df-dc 782 This theorem is referenced by:  bj-d0clsepcl  12124
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