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Theorem pm5.14dc 911
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 835 . 2 (DECID 𝜓 ↔ (𝜓 ∨ ¬ 𝜓))
2 ax-1 6 . . 3 (𝜓 → (𝜑𝜓))
3 ax-in2 615 . . 3 𝜓 → (𝜓𝜒))
42, 3orim12i 759 . 2 ((𝜓 ∨ ¬ 𝜓) → ((𝜑𝜓) ∨ (𝜓𝜒)))
51, 4sylbi 121 1 (DECID 𝜓 → ((𝜑𝜓) ∨ (𝜓𝜒)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wo 708  DECID wdc 834
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in2 615  ax-io 709
This theorem depends on definitions:  df-bi 117  df-dc 835
This theorem is referenced by:  pm5.13dc  912
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