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Theorem pm2.25dc 826
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 777 . 2 (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑))
2 orel1 677 . . 3 𝜑 → ((𝜑𝜓) → 𝜓))
32orim2i 711 . 2 ((𝜑 ∨ ¬ 𝜑) → (𝜑 ∨ ((𝜑𝜓) → 𝜓)))
41, 3sylbi 119 1 (DECID 𝜑 → (𝜑 ∨ ((𝜑𝜓) → 𝜓)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wo 662  DECID wdc 776
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-in2 578  ax-io 663
This theorem depends on definitions:  df-bi 115  df-dc 777
This theorem is referenced by: (None)
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