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Theorem pm5.12dc 816
Description: Excluded middle with antecedents for a decidable consequent. Based on theorem *5.12 of [WhiteheadRussell] p. 123. (Contributed by Jim Kingdon, 30-Mar-2018.)
Assertion
Ref Expression
pm5.12dc (DECID 𝜓 → ((𝜑𝜓) ∨ (𝜑 → ¬ 𝜓)))

Proof of Theorem pm5.12dc
StepHypRef Expression
1 df-dc 743 . 2 (DECID 𝜓 ↔ (𝜓 ∨ ¬ 𝜓))
2 ax-1 5 . . 3 (𝜓 → (𝜑𝜓))
3 ax-1 5 . . 3 𝜓 → (𝜑 → ¬ 𝜓))
42, 3orim12i 676 . 2 ((𝜓 ∨ ¬ 𝜓) → ((𝜑𝜓) ∨ (𝜑 → ¬ 𝜓)))
51, 4sylbi 114 1 (DECID 𝜓 → ((𝜑𝜓) ∨ (𝜑 → ¬ 𝜓)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wo 629  DECID wdc 742
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630
This theorem depends on definitions:  df-bi 110  df-dc 743
This theorem is referenced by: (None)
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