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Theorem pm3.12dc 925
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 924 . . . 4 (DECID 𝜑 → (DECID 𝜓 → (¬ (¬ 𝜑 ∨ ¬ 𝜓) → (𝜑𝜓))))
21imp 123 . . 3 ((DECID 𝜑DECID 𝜓) → (¬ (¬ 𝜑 ∨ ¬ 𝜓) → (𝜑𝜓)))
3 dcn 810 . . . . . 6 (DECID 𝜑DECID ¬ 𝜑)
4 dcn 810 . . . . . 6 (DECID 𝜓DECID ¬ 𝜓)
5 dcor 902 . . . . . 6 (DECID ¬ 𝜑 → (DECID ¬ 𝜓DECID𝜑 ∨ ¬ 𝜓)))
63, 4, 5syl2im 38 . . . . 5 (DECID 𝜑 → (DECID 𝜓DECID𝜑 ∨ ¬ 𝜓)))
7 dfordc 860 . . . . 5 (DECID𝜑 ∨ ¬ 𝜓) → (((¬ 𝜑 ∨ ¬ 𝜓) ∨ (𝜑𝜓)) ↔ (¬ (¬ 𝜑 ∨ ¬ 𝜓) → (𝜑𝜓))))
86, 7syl6 33 . . . 4 (DECID 𝜑 → (DECID 𝜓 → (((¬ 𝜑 ∨ ¬ 𝜓) ∨ (𝜑𝜓)) ↔ (¬ (¬ 𝜑 ∨ ¬ 𝜓) → (𝜑𝜓)))))
98imp 123 . . 3 ((DECID 𝜑DECID 𝜓) → (((¬ 𝜑 ∨ ¬ 𝜓) ∨ (𝜑𝜓)) ↔ (¬ (¬ 𝜑 ∨ ¬ 𝜓) → (𝜑𝜓))))
102, 9mpbird 166 . 2 ((DECID 𝜑DECID 𝜓) → ((¬ 𝜑 ∨ ¬ 𝜓) ∨ (𝜑𝜓)))
1110ex 114 1 (DECID 𝜑 → (DECID 𝜓 → ((¬ 𝜑 ∨ ¬ 𝜓) ∨ (𝜑𝜓))))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wb 104  wo 680  DECID wdc 802
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 586  ax-in2 587  ax-io 681
This theorem depends on definitions:  df-bi 116  df-stab 799  df-dc 803
This theorem is referenced by: (None)
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