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Theorem pm5.11dc 853
Description: A decidable proposition or its negation implies a second proposition. Based on theorem *5.11 of [WhiteheadRussell] p. 123. (Contributed by Jim Kingdon, 29-Mar-2018.)
Assertion
Ref Expression
pm5.11dc (DECID 𝜑 → (DECID 𝜓 → ((𝜑𝜓) ∨ (¬ 𝜑𝜓))))

Proof of Theorem pm5.11dc
StepHypRef Expression
1 dcim 822 . 2 (DECID 𝜑 → (DECID 𝜓DECID (𝜑𝜓)))
2 pm2.5dc 801 . . 3 (DECID 𝜑 → (¬ (𝜑𝜓) → (¬ 𝜑𝜓)))
3 pm2.54dc 828 . . 3 (DECID (𝜑𝜓) → ((¬ (𝜑𝜓) → (¬ 𝜑𝜓)) → ((𝜑𝜓) ∨ (¬ 𝜑𝜓))))
42, 3syl5com 29 . 2 (DECID 𝜑 → (DECID (𝜑𝜓) → ((𝜑𝜓) ∨ (¬ 𝜑𝜓))))
51, 4syld 44 1 (DECID 𝜑 → (DECID 𝜓 → ((𝜑𝜓) ∨ (¬ 𝜑𝜓))))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  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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