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Theorem dcbii 758
Description: The equivalent of a decidable proposition is decidable. (Contributed by Jim Kingdon, 28-Mar-2018.)
Hypothesis
Ref Expression
dcbii.1 (𝜑𝜓)
Assertion
Ref Expression
dcbii (DECID 𝜑DECID 𝜓)

Proof of Theorem dcbii
StepHypRef Expression
1 dcbii.1 . . 3 (𝜑𝜓)
21notbii 604 . . 3 𝜑 ↔ ¬ 𝜓)
31, 2orbi12i 691 . 2 ((𝜑 ∨ ¬ 𝜑) ↔ (𝜓 ∨ ¬ 𝜓))
4 df-dc 754 . 2 (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑))
5 df-dc 754 . 2 (DECID 𝜓 ↔ (𝜓 ∨ ¬ 𝜓))
63, 4, 53bitr4i 205 1 (DECID 𝜑DECID 𝜓)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  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:  dcbi  855  dcned  2226  euxfr2dc  2749
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