ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  pm5.54dc GIF version

Theorem pm5.54dc 865
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 781 . . 3 (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑))
2 simpr 108 . . . . 5 ((𝜑𝜓) → 𝜓)
3 ax-ia3 106 . . . . 5 (𝜑 → (𝜓 → (𝜑𝜓)))
42, 3impbid2 141 . . . 4 (𝜑 → ((𝜑𝜓) ↔ 𝜓))
5 simpl 107 . . . . 5 ((𝜑𝜓) → 𝜑)
6 ax-in2 580 . . . . 5 𝜑 → (𝜑 → (𝜑𝜓)))
75, 6impbid2 141 . . . 4 𝜑 → ((𝜑𝜓) ↔ 𝜑))
84, 7orim12i 711 . . 3 ((𝜑 ∨ ¬ 𝜑) → (((𝜑𝜓) ↔ 𝜓) ∨ ((𝜑𝜓) ↔ 𝜑)))
91, 8sylbi 119 . 2 (DECID 𝜑 → (((𝜑𝜓) ↔ 𝜓) ∨ ((𝜑𝜓) ↔ 𝜑)))
109orcomd 683 1 (DECID 𝜑 → (((𝜑𝜓) ↔ 𝜑) ∨ ((𝜑𝜓) ↔ 𝜓)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  wb 103  wo 664  DECID wdc 780
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 580  ax-io 665
This theorem depends on definitions:  df-bi 115  df-dc 781
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator