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Theorem pm5.24dc 1377
 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 1376 . . . . 5 (DECID 𝜑 → (DECID 𝜓 → ((𝜑𝜓) ↔ ((𝜑𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓)))))
21imp 123 . . . 4 ((DECID 𝜑DECID 𝜓) → ((𝜑𝜓) ↔ ((𝜑𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓))))
32notbid 657 . . 3 ((DECID 𝜑DECID 𝜓) → (¬ (𝜑𝜓) ↔ ¬ ((𝜑𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓))))
4 xordc 1371 . . . 4 (DECID 𝜑 → (DECID 𝜓 → (¬ (𝜑𝜓) ↔ ((𝜑 ∧ ¬ 𝜓) ∨ (𝜓 ∧ ¬ 𝜑)))))
54imp 123 . . 3 ((DECID 𝜑DECID 𝜓) → (¬ (𝜑𝜓) ↔ ((𝜑 ∧ ¬ 𝜓) ∨ (𝜓 ∧ ¬ 𝜑))))
63, 5bitr3d 189 . 2 ((DECID 𝜑DECID 𝜓) → (¬ ((𝜑𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓)) ↔ ((𝜑 ∧ ¬ 𝜓) ∨ (𝜓 ∧ ¬ 𝜑))))
76ex 114 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 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-in1 604  ax-in2 605  ax-io 699 This theorem depends on definitions:  df-bi 116  df-stab 817  df-dc 821  df-xor 1355 This theorem is referenced by: (None)
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