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Theorem pm3.13dc 905
Description: Theorem *3.13 of [WhiteheadRussell] p. 111, but for decidable propositions. The converse, pm3.14 705, holds for all propositions. (Contributed by Jim Kingdon, 22-Apr-2018.)
Assertion
Ref Expression
pm3.13dc (DECID 𝜑 → (DECID 𝜓 → (¬ (𝜑𝜓) → (¬ 𝜑 ∨ ¬ 𝜓))))

Proof of Theorem pm3.13dc
StepHypRef Expression
1 dcn 784 . . 3 (DECID 𝜑DECID ¬ 𝜑)
2 dcn 784 . . 3 (DECID 𝜓DECID ¬ 𝜓)
3 dcor 881 . . 3 (DECID ¬ 𝜑 → (DECID ¬ 𝜓DECID𝜑 ∨ ¬ 𝜓)))
41, 2, 3syl2im 38 . 2 (DECID 𝜑 → (DECID 𝜓DECID𝜑 ∨ ¬ 𝜓)))
5 pm3.11dc 903 . 2 (DECID 𝜑 → (DECID 𝜓 → (¬ (¬ 𝜑 ∨ ¬ 𝜓) → (𝜑𝜓))))
6 con1dc 791 . 2 (DECID𝜑 ∨ ¬ 𝜓) → ((¬ (¬ 𝜑 ∨ ¬ 𝜓) → (𝜑𝜓)) → (¬ (𝜑𝜓) → (¬ 𝜑 ∨ ¬ 𝜓))))
74, 5, 6syl6c 65 1 (DECID 𝜑 → (DECID 𝜓 → (¬ (𝜑𝜓) → (¬ 𝜑 ∨ ¬ 𝜓))))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  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-in1 579  ax-in2 580  ax-io 665
This theorem depends on definitions:  df-bi 115  df-dc 781
This theorem is referenced by: (None)
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