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Theorem dcbid 747
Description: The equivalent of a decidable proposition is decidable. (Contributed by Jim Kingdon, 7-Sep-2019.)
Hypothesis
Ref Expression
dcbid.1 (φ → (ψχ))
Assertion
Ref Expression
dcbid (φ → (DECID ψDECID χ))

Proof of Theorem dcbid
StepHypRef Expression
1 dcbid.1 . . 3 (φ → (ψχ))
21notbid 591 . . 3 (φ → (¬ ψ ↔ ¬ χ))
31, 2orbi12d 706 . 2 (φ → ((ψ ¬ ψ) ↔ (χ ¬ χ)))
4 df-dc 742 . 2 (DECID ψ ↔ (ψ ¬ ψ))
5 df-dc 742 . 2 (DECID χ ↔ (χ ¬ χ))
63, 4, 53bitr4g 212 1 (φ → (DECID ψDECID χ))
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:  ltdcpi  6307  enqdc  6345  enqdc1  6346  ltdcnq  6381
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