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Theorem pm4.83dc 951
Description: Theorem *4.83 of [WhiteheadRussell] p. 122, for decidable propositions. As with other case elimination theorems, like pm2.61dc 865, it only holds for decidable propositions. (Contributed by Jim Kingdon, 12-May-2018.)
Assertion
Ref Expression
pm4.83dc (DECID 𝜑 → (((𝜑𝜓) ∧ (¬ 𝜑𝜓)) ↔ 𝜓))

Proof of Theorem pm4.83dc
StepHypRef Expression
1 df-dc 835 . . 3 (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑))
2 pm3.44 715 . . . 4 (((𝜑𝜓) ∧ (¬ 𝜑𝜓)) → ((𝜑 ∨ ¬ 𝜑) → 𝜓))
32com12 30 . . 3 ((𝜑 ∨ ¬ 𝜑) → (((𝜑𝜓) ∧ (¬ 𝜑𝜓)) → 𝜓))
41, 3sylbi 121 . 2 (DECID 𝜑 → (((𝜑𝜓) ∧ (¬ 𝜑𝜓)) → 𝜓))
5 ax-1 6 . . 3 (𝜓 → (𝜑𝜓))
6 ax-1 6 . . 3 (𝜓 → (¬ 𝜑𝜓))
75, 6jca 306 . 2 (𝜓 → ((𝜑𝜓) ∧ (¬ 𝜑𝜓)))
84, 7impbid1 142 1 (DECID 𝜑 → (((𝜑𝜓) ∧ (¬ 𝜑𝜓)) ↔ 𝜓))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  wb 105  wo 708  DECID wdc 834
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709
This theorem depends on definitions:  df-bi 117  df-dc 835
This theorem is referenced by: (None)
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