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Theorem pm2.61ddc 847
Description: Deduction eliminating a decidable antecedent. (Contributed by Jim Kingdon, 4-May-2018.)
Hypotheses
Ref Expression
pm2.61ddc.1 (𝜑 → (𝜓𝜒))
pm2.61ddc.2 (𝜑 → (¬ 𝜓𝜒))
Assertion
Ref Expression
pm2.61ddc (DECID 𝜓 → (𝜑𝜒))

Proof of Theorem pm2.61ddc
StepHypRef Expression
1 df-dc 821 . 2 (DECID 𝜓 ↔ (𝜓 ∨ ¬ 𝜓))
2 pm2.61ddc.1 . . . 4 (𝜑 → (𝜓𝜒))
32com12 30 . . 3 (𝜓 → (𝜑𝜒))
4 pm2.61ddc.2 . . . 4 (𝜑 → (¬ 𝜓𝜒))
54com12 30 . . 3 𝜓 → (𝜑𝜒))
63, 5jaoi 706 . 2 ((𝜓 ∨ ¬ 𝜓) → (𝜑𝜒))
71, 6sylbi 120 1 (DECID 𝜓 → (𝜑𝜒))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wo 698  DECID wdc 820
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699
This theorem depends on definitions:  df-bi 116  df-dc 821
This theorem is referenced by:  bijadc  868
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