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Theorem pm5.14dc 855
Description: A decidable proposition is implied by or implies other propositions. Based on theorem *5.14 of [WhiteheadRussell] p. 123. (Contributed by Jim Kingdon, 30-Mar-2018.)
Assertion
Ref Expression
pm5.14dc (DECID 𝜓 → ((𝜑𝜓) ∨ (𝜓𝜒)))

Proof of Theorem pm5.14dc
StepHypRef Expression
1 df-dc 781 . 2 (DECID 𝜓 ↔ (𝜓 ∨ ¬ 𝜓))
2 ax-1 5 . . 3 (𝜓 → (𝜑𝜓))
3 ax-in2 580 . . 3 𝜓 → (𝜓𝜒))
42, 3orim12i 711 . 2 ((𝜓 ∨ ¬ 𝜓) → ((𝜑𝜓) ∨ (𝜓𝜒)))
51, 4sylbi 119 1 (DECID 𝜓 → ((𝜑𝜓) ∨ (𝜓𝜒)))
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-in2 580  ax-io 665
This theorem depends on definitions:  df-bi 115  df-dc 781
This theorem is referenced by:  pm5.13dc  856
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