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Theorem pm5.54dc 904
 Description: A conjunction is equivalent to one of its conjuncts, given a decidable conjunct. Based on theorem *5.54 of [WhiteheadRussell] p. 125. (Contributed by Jim Kingdon, 30-Mar-2018.)
Assertion
Ref Expression
pm5.54dc (DECID 𝜑 → (((𝜑𝜓) ↔ 𝜑) ∨ ((𝜑𝜓) ↔ 𝜓)))

Proof of Theorem pm5.54dc
StepHypRef Expression
1 df-dc 821 . . 3 (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑))
2 simpr 109 . . . . 5 ((𝜑𝜓) → 𝜓)
3 ax-ia3 107 . . . . 5 (𝜑 → (𝜓 → (𝜑𝜓)))
42, 3impbid2 142 . . . 4 (𝜑 → ((𝜑𝜓) ↔ 𝜓))
5 simpl 108 . . . . 5 ((𝜑𝜓) → 𝜑)
6 ax-in2 605 . . . . 5 𝜑 → (𝜑 → (𝜑𝜓)))
75, 6impbid2 142 . . . 4 𝜑 → ((𝜑𝜓) ↔ 𝜑))
84, 7orim12i 749 . . 3 ((𝜑 ∨ ¬ 𝜑) → (((𝜑𝜓) ↔ 𝜓) ∨ ((𝜑𝜓) ↔ 𝜑)))
91, 8sylbi 120 . 2 (DECID 𝜑 → (((𝜑𝜓) ↔ 𝜓) ∨ ((𝜑𝜓) ↔ 𝜑)))
109orcomd 719 1 (DECID 𝜑 → (((𝜑𝜓) ↔ 𝜑) ∨ ((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-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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