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Theorem dfor2dc 865
Description: Disjunction expressed in terms of implication only, for a decidable proposition. Based on theorem *5.25 of [WhiteheadRussell] p. 124. (Contributed by Jim Kingdon, 27-Mar-2018.)
Assertion
Ref Expression
dfor2dc (DECID 𝜑 → ((𝜑𝜓) ↔ ((𝜑𝜓) → 𝜓)))

Proof of Theorem dfor2dc
StepHypRef Expression
1 pm2.62 722 . 2 ((𝜑𝜓) → ((𝜑𝜓) → 𝜓))
2 pm2.68dc 864 . 2 (DECID 𝜑 → (((𝜑𝜓) → 𝜓) → (𝜑𝜓)))
31, 2impbid2 142 1 (DECID 𝜑 → ((𝜑𝜓) ↔ ((𝜑𝜓) → 𝜓)))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104  wo 682  DECID wdc 804
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683
This theorem depends on definitions:  df-bi 116  df-dc 805
This theorem is referenced by:  imimorbdc  866
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