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Theorem pm2.54dc 801
 Description: Deriving disjunction from implication for a decidable proposition. Based on theorem *2.54 of [WhiteheadRussell] p. 107. The converse, pm2.53 651, 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 757 . 2 (DECID 𝜑DECID ¬ 𝜑)
2 notnotrdc 762 . . . . 5 (DECID 𝜑 → (¬ ¬ 𝜑𝜑))
3 orc 643 . . . . 5 (𝜑 → (𝜑𝜓))
42, 3syl6 33 . . . 4 (DECID 𝜑 → (¬ ¬ 𝜑 → (𝜑𝜓)))
54a1d 22 . . 3 (DECID 𝜑 → (DECID ¬ 𝜑 → (¬ ¬ 𝜑 → (𝜑𝜓))))
6 olc 642 . . . 4 (𝜓 → (𝜑𝜓))
76a1i 9 . . 3 (DECID 𝜑 → (𝜓 → (𝜑𝜓)))
85, 7jaddc 772 . 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 639  DECID wdc 753 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640 This theorem depends on definitions:  df-bi 114  df-dc 754 This theorem is referenced by:  dfordc  802  pm2.68dc  804  pm4.79dc  820  pm5.11dc  826
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