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Theorem pm2.25dc 879
Description: Elimination of disjunction based on a disjunction, for a decidable proposition. Based on theorem *2.25 of [WhiteheadRussell] p. 104. (Contributed by NM, 3-Jan-2005.)
Assertion
Ref Expression
pm2.25dc (DECID 𝜑 → (𝜑 ∨ ((𝜑𝜓) → 𝜓)))

Proof of Theorem pm2.25dc
StepHypRef Expression
1 df-dc 821 . 2 (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑))
2 orel1 715 . . 3 𝜑 → ((𝜑𝜓) → 𝜓))
32orim2i 751 . 2 ((𝜑 ∨ ¬ 𝜑) → (𝜑 ∨ ((𝜑𝜓) → 𝜓)))
41, 3sylbi 120 1 (DECID 𝜑 → (𝜑 ∨ ((𝜑𝜓) → 𝜓)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wo 698  DECID wdc 820
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-in2 605  ax-io 699
This theorem depends on definitions:  df-bi 116  df-dc 821
This theorem is referenced by: (None)
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