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Theorem bj-dcbi 9359
Description: Equivalence property for DECID. TODO: solve conflict with dcbi 843; minimize dcbii 746 and dcbid 747 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 9356 . . 3 ((φψ) → (¬ φ ↔ ¬ ψ))
31, 2orbi12d 706 . 2 ((φψ) → ((φ ¬ φ) ↔ (ψ ¬ ψ)))
4 df-dc 742 . 2 (DECID φ ↔ (φ ¬ φ))
5 df-dc 742 . 2 (DECID ψ ↔ (ψ ¬ ψ))
63, 4, 53bitr4g 212 1 ((φψ) → (DECID φDECID ψ))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 98   wo 628  DECID wdc 741
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 629
This theorem depends on definitions:  df-bi 110  df-dc 742
This theorem is referenced by:  bj-d0clsepcl  9360
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