ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  pm5.24dc GIF version

Theorem pm5.24dc 1305
Description: Theorem *5.24 of [WhiteheadRussell] p. 124, but for decidable propositions. (Contributed by Jim Kingdon, 5-May-2018.)
Assertion
Ref Expression
pm5.24dc (DECID 𝜑 → (DECID 𝜓 → (¬ ((𝜑𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓)) ↔ ((𝜑 ∧ ¬ 𝜓) ∨ (𝜓 ∧ ¬ 𝜑)))))

Proof of Theorem pm5.24dc
StepHypRef Expression
1 dfbi3dc 1304 . . . . 5 (DECID 𝜑 → (DECID 𝜓 → ((𝜑𝜓) ↔ ((𝜑𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓)))))
21imp 119 . . . 4 ((DECID 𝜑DECID 𝜓) → ((𝜑𝜓) ↔ ((𝜑𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓))))
32notbid 602 . . 3 ((DECID 𝜑DECID 𝜓) → (¬ (𝜑𝜓) ↔ ¬ ((𝜑𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓))))
4 xordc 1299 . . . 4 (DECID 𝜑 → (DECID 𝜓 → (¬ (𝜑𝜓) ↔ ((𝜑 ∧ ¬ 𝜓) ∨ (𝜓 ∧ ¬ 𝜑)))))
54imp 119 . . 3 ((DECID 𝜑DECID 𝜓) → (¬ (𝜑𝜓) ↔ ((𝜑 ∧ ¬ 𝜓) ∨ (𝜓 ∧ ¬ 𝜑))))
63, 5bitr3d 183 . 2 ((DECID 𝜑DECID 𝜓) → (¬ ((𝜑𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓)) ↔ ((𝜑 ∧ ¬ 𝜓) ∨ (𝜓 ∧ ¬ 𝜑))))
76ex 112 1 (DECID 𝜑 → (DECID 𝜓 → (¬ ((𝜑𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓)) ↔ ((𝜑 ∧ ¬ 𝜓) ∨ (𝜓 ∧ ¬ 𝜑)))))
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  df-xor 1283
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator