ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  pm3.12dc GIF version

Theorem pm3.12dc 876
Description: Theorem *3.12 of [WhiteheadRussell] p. 111, but for decidable propositions. (Contributed by Jim Kingdon, 22-Apr-2018.)
Assertion
Ref Expression
pm3.12dc (DECID 𝜑 → (DECID 𝜓 → ((¬ 𝜑 ∨ ¬ 𝜓) ∨ (𝜑𝜓))))

Proof of Theorem pm3.12dc
StepHypRef Expression
1 pm3.11dc 875 . . . 4 (DECID 𝜑 → (DECID 𝜓 → (¬ (¬ 𝜑 ∨ ¬ 𝜓) → (𝜑𝜓))))
21imp 119 . . 3 ((DECID 𝜑DECID 𝜓) → (¬ (¬ 𝜑 ∨ ¬ 𝜓) → (𝜑𝜓)))
3 dcn 757 . . . . . 6 (DECID 𝜑DECID ¬ 𝜑)
4 dcn 757 . . . . . 6 (DECID 𝜓DECID ¬ 𝜓)
5 dcor 854 . . . . . 6 (DECID ¬ 𝜑 → (DECID ¬ 𝜓DECID𝜑 ∨ ¬ 𝜓)))
63, 4, 5syl2im 38 . . . . 5 (DECID 𝜑 → (DECID 𝜓DECID𝜑 ∨ ¬ 𝜓)))
7 dfordc 802 . . . . 5 (DECID𝜑 ∨ ¬ 𝜓) → (((¬ 𝜑 ∨ ¬ 𝜓) ∨ (𝜑𝜓)) ↔ (¬ (¬ 𝜑 ∨ ¬ 𝜓) → (𝜑𝜓))))
86, 7syl6 33 . . . 4 (DECID 𝜑 → (DECID 𝜓 → (((¬ 𝜑 ∨ ¬ 𝜓) ∨ (𝜑𝜓)) ↔ (¬ (¬ 𝜑 ∨ ¬ 𝜓) → (𝜑𝜓)))))
98imp 119 . . 3 ((DECID 𝜑DECID 𝜓) → (((¬ 𝜑 ∨ ¬ 𝜓) ∨ (𝜑𝜓)) ↔ (¬ (¬ 𝜑 ∨ ¬ 𝜓) → (𝜑𝜓))))
102, 9mpbird 160 . 2 ((DECID 𝜑DECID 𝜓) → ((¬ 𝜑 ∨ ¬ 𝜓) ∨ (𝜑𝜓)))
1110ex 112 1 (DECID 𝜑 → (DECID 𝜓 → ((¬ 𝜑 ∨ ¬ 𝜓) ∨ (𝜑𝜓))))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 101  wb 102  wo 639  DECID wdc 753
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640
This theorem depends on definitions:  df-bi 114  df-dc 754
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator