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Theorem dcbi 843
Description: An equivalence of two decidable propositions is decidable. (Contributed by Jim Kingdon, 12-Apr-2018.)
Assertion
Ref Expression
dcbi (DECID φ → (DECID ψDECID (φψ)))

Proof of Theorem dcbi
StepHypRef Expression
1 dcim 783 . . 3 (DECID φ → (DECID ψDECID (φψ)))
2 dcim 783 . . . 4 (DECID ψ → (DECID φDECID (ψφ)))
32com12 27 . . 3 (DECID φ → (DECID ψDECID (ψφ)))
4 dcan 841 . . 3 (DECID (φψ) → (DECID (ψφ) → DECID ((φψ) (ψφ))))
51, 3, 4syl6c 60 . 2 (DECID φ → (DECID ψDECID ((φψ) (ψφ))))
6 dfbi2 368 . . 3 ((φψ) ↔ ((φψ) (ψφ)))
76dcbii 746 . 2 (DECID (φψ) ↔ DECID ((φψ) (ψφ)))
85, 7syl6ibr 151 1 (DECID φ → (DECID ψDECID (φψ)))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98  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:  xor3dc  1275  pm5.15dc  1277  bilukdc  1284  xordidc  1287
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