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Theorem pm2.54dc 826
 Description: Deriving disjunction from implication for a decidable proposition. Based on theorem *2.54 of [WhiteheadRussell] p. 107. The converse, pm2.53 674, holds whether the proposition is decidable or not. (Contributed by Jim Kingdon, 26-Mar-2018.)
Assertion
Ref Expression
pm2.54dc (DECID 𝜑 → ((¬ 𝜑𝜓) → (𝜑𝜓)))

Proof of Theorem pm2.54dc
StepHypRef Expression
1 dcn 782 . 2 (DECID 𝜑DECID ¬ 𝜑)
2 notnotrdc 787 . . . . 5 (DECID 𝜑 → (¬ ¬ 𝜑𝜑))
3 orc 666 . . . . 5 (𝜑 → (𝜑𝜓))
42, 3syl6 33 . . . 4 (DECID 𝜑 → (¬ ¬ 𝜑 → (𝜑𝜓)))
54a1d 22 . . 3 (DECID 𝜑 → (DECID ¬ 𝜑 → (¬ ¬ 𝜑 → (𝜑𝜓))))
6 olc 665 . . . 4 (𝜓 → (𝜑𝜓))
76a1i 9 . . 3 (DECID 𝜑 → (𝜓 → (𝜑𝜓)))
85, 7jaddc 797 . 2 (DECID 𝜑 → (DECID ¬ 𝜑 → ((¬ 𝜑𝜓) → (𝜑𝜓))))
91, 8mpd 13 1 (DECID 𝜑 → ((¬ 𝜑𝜓) → (𝜑𝜓)))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 662  DECID wdc 778 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663 This theorem depends on definitions:  df-bi 115  df-dc 779 This theorem is referenced by:  dfordc  827  pm2.68dc  829  pm4.79dc  845  pm5.11dc  851
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