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Theorem dcbid 759
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 602 . . 3 (𝜑 → (¬ 𝜓 ↔ ¬ 𝜒))
31, 2orbi12d 717 . 2 (𝜑 → ((𝜓 ∨ ¬ 𝜓) ↔ (𝜒 ∨ ¬ 𝜒)))
4 df-dc 754 . 2 (DECID 𝜓 ↔ (𝜓 ∨ ¬ 𝜓))
5 df-dc 754 . 2 (DECID 𝜒 ↔ (𝜒 ∨ ¬ 𝜒))
63, 4, 53bitr4g 216 1 (𝜑 → (DECID 𝜓DECID 𝜒))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 102  wo 639  DECID wdc 753
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
This theorem is referenced by:  ltdcpi  6478  enqdc  6516  enqdc1  6517  ltdcnq  6552  dvdsdc  10115
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