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Theorem pm5.11dc 815
 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 784 . 2 (DECID 𝜑 → (DECID 𝜓DECID (𝜑𝜓)))
2 pm2.5dc 763 . . 3 (DECID 𝜑 → (¬ (𝜑𝜓) → (¬ 𝜑𝜓)))
3 pm2.54dc 790 . . 3 (DECID (𝜑𝜓) → ((¬ (𝜑𝜓) → (¬ 𝜑𝜓)) → ((𝜑𝜓) ∨ (¬ 𝜑𝜓))))
42, 3syl5com 26 . 2 (DECID 𝜑 → (DECID (𝜑𝜓) → ((𝜑𝜓) ∨ (¬ 𝜑𝜓))))
51, 4syld 40 1 (DECID 𝜑 → (DECID 𝜓 → ((𝜑𝜓) ∨ (¬ 𝜑𝜓))))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 629  DECID wdc 742 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630 This theorem depends on definitions:  df-bi 110  df-dc 743 This theorem is referenced by: (None)
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