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Theorem dcim 783
Description: An implication between two decidable propositions is decidable. (Contributed by Jim Kingdon, 28-Mar-2018.)
Assertion
Ref Expression
dcim (DECID φ → (DECID ψDECID (φψ)))

Proof of Theorem dcim
StepHypRef Expression
1 df-dc 742 . 2 (DECID φ ↔ (φ ¬ φ))
2 df-dc 742 . . . . . . . 8 (DECID ψ ↔ (ψ ¬ ψ))
32anbi2i 430 . . . . . . 7 ((φ DECID ψ) ↔ (φ (ψ ¬ ψ)))
4 andi 730 . . . . . . 7 ((φ (ψ ¬ ψ)) ↔ ((φ ψ) (φ ¬ ψ)))
53, 4bitri 173 . . . . . 6 ((φ DECID ψ) ↔ ((φ ψ) (φ ¬ ψ)))
6 pm3.4 316 . . . . . . 7 ((φ ψ) → (φψ))
7 annimim 781 . . . . . . 7 ((φ ¬ ψ) → ¬ (φψ))
86, 7orim12i 675 . . . . . 6 (((φ ψ) (φ ¬ ψ)) → ((φψ) ¬ (φψ)))
95, 8sylbi 114 . . . . 5 ((φ DECID ψ) → ((φψ) ¬ (φψ)))
10 df-dc 742 . . . . 5 (DECID (φψ) ↔ ((φψ) ¬ (φψ)))
119, 10sylibr 137 . . . 4 ((φ DECID ψ) → DECID (φψ))
1211ex 108 . . 3 (φ → (DECID ψDECID (φψ)))
13 ax-in2 545 . . . . 5 φ → (φψ))
1413a1d 22 . . . 4 φ → (DECID ψ → (φψ)))
15 orc 632 . . . . 5 ((φψ) → ((φψ) ¬ (φψ)))
1615, 10sylibr 137 . . . 4 ((φψ) → DECID (φψ))
1714, 16syl6 29 . . 3 φ → (DECID ψDECID (φψ)))
1812, 17jaoi 635 . 2 ((φ ¬ φ) → (DECID ψDECID (φψ)))
191, 18sylbi 114 1 (DECID φ → (DECID ψDECID (φψ)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wa 97   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:  pm4.79dc  808  pm5.11dc  814  dcbi  843  annimdc  844
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