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Theorem dcim 777
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 715 . 2 (DECID φ ↔ (φ ¬ φ))
2 df-dc 715 . . . . . . . 8 (DECID ψ ↔ (ψ ¬ ψ))
32anbi2i 429 . . . . . . 7 ((φ DECID ψ) ↔ (φ (ψ ¬ ψ)))
4 andi 706 . . . . . . 7 ((φ (ψ ¬ ψ)) ↔ ((φ ψ) (φ ¬ ψ)))
53, 4bitri 171 . . . . . 6 ((φ DECID ψ) ↔ ((φ ψ) (φ ¬ ψ)))
6 pm3.4 314 . . . . . . 7 ((φ ψ) → (φψ))
7 annimim 776 . . . . . . 7 ((φ ¬ ψ) → ¬ (φψ))
86, 7orim12i 651 . . . . . 6 (((φ ψ) (φ ¬ ψ)) → ((φψ) ¬ (φψ)))
95, 8sylbi 112 . . . . 5 ((φ DECID ψ) → ((φψ) ¬ (φψ)))
10 df-dc 715 . . . . 5 (DECID (φψ) ↔ ((φψ) ¬ (φψ)))
119, 10sylibr 135 . . . 4 ((φ DECID ψ) → DECID (φψ))
1211ex 106 . . 3 (φ → (DECID ψDECID (φψ)))
13 ax-in2 527 . . . . 5 φ → (φψ))
1413a1d 20 . . . 4 φ → (DECID ψ → (φψ)))
15 orc 609 . . . . 5 ((φψ) → ((φψ) ¬ (φψ)))
1615, 10sylibr 135 . . . 4 ((φψ) → DECID (φψ))
1714, 16syl6 27 . . 3 φ → (DECID ψDECID (φψ)))
1812, 17jaoi 612 . 2 ((φ ¬ φ) → (DECID ψDECID (φψ)))
191, 18sylbi 112 1 (DECID φ → (DECID ψDECID (φψ)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wa 95   wo 605  DECID wdc 714
This theorem is referenced by:  pm4.79dc  798  pm5.11dc  803  dcbi  821
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 97  ax-ia2 98  ax-ia3 99  ax-in1 526  ax-in2 527  ax-io 606
This theorem depends on definitions:  df-bi 108  df-dc 715
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