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Theorem pm4.83dc 936
Description: Theorem *4.83 of [WhiteheadRussell] p. 122, for decidable propositions. As with other case elimination theorems, like pm2.61dc 851, 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 821 . . 3 (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑))
2 pm3.44 705 . . . 4 (((𝜑𝜓) ∧ (¬ 𝜑𝜓)) → ((𝜑 ∨ ¬ 𝜑) → 𝜓))
32com12 30 . . 3 ((𝜑 ∨ ¬ 𝜑) → (((𝜑𝜓) ∧ (¬ 𝜑𝜓)) → 𝜓))
41, 3sylbi 120 . 2 (DECID 𝜑 → (((𝜑𝜓) ∧ (¬ 𝜑𝜓)) → 𝜓))
5 ax-1 6 . . 3 (𝜓 → (𝜑𝜓))
6 ax-1 6 . . 3 (𝜓 → (¬ 𝜑𝜓))
75, 6jca 304 . 2 (𝜓 → ((𝜑𝜓) ∧ (¬ 𝜑𝜓)))
84, 7impbid1 141 1 (DECID 𝜑 → (((𝜑𝜓) ∧ (¬ 𝜑𝜓)) ↔ 𝜓))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wb 104  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-io 699
This theorem depends on definitions:  df-bi 116  df-dc 821
This theorem is referenced by: (None)
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