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Theorem pm2.54dc 789
Description: Deriving disjunction from implication for a decidable proposition. Based on theorem *2.54 of [WhiteheadRussell] p. 107. The converse, pm2.53 640, holds whether the proposition is decidable or not. (Contributed by Jim Kingdon, 26-Mar-2018.)
Assertion
Ref Expression
pm2.54dc (DECID φ → ((¬ φψ) → (φ ψ)))

Proof of Theorem pm2.54dc
StepHypRef Expression
1 dcn 745 . 2 (DECID φDECID ¬ φ)
2 notnot2dc 750 . . . . 5 (DECID φ → (¬ ¬ φφ))
3 orc 632 . . . . 5 (φ → (φ ψ))
42, 3syl6 29 . . . 4 (DECID φ → (¬ ¬ φ → (φ ψ)))
54a1d 22 . . 3 (DECID φ → (DECID ¬ φ → (¬ ¬ φ → (φ ψ))))
6 olc 631 . . . 4 (ψ → (φ ψ))
76a1i 9 . . 3 (DECID φ → (ψ → (φ ψ)))
85, 7jaddc 760 . 2 (DECID φ → (DECID ¬ φ → ((¬ φψ) → (φ ψ))))
91, 8mpd 13 1 (DECID φ → ((¬ φψ) → (φ ψ)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   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:  dfordc  790  pm2.68dc  792  pm4.79dc  808  pm5.11dc  814
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