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Theorem pm2.61ddc 751
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 734 . 2 (DECID ψ ↔ (ψ ¬ ψ))
2 pm2.61ddc.1 . . . 4 (φ → (ψχ))
32com12 27 . . 3 (ψ → (φχ))
4 pm2.61ddc.2 . . . 4 (φ → (¬ ψχ))
54com12 27 . . 3 ψ → (φχ))
63, 5jaoi 623 . 2 ((ψ ¬ ψ) → (φχ))
71, 6sylbi 114 1 (DECID ψ → (φχ))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wo 616  DECID wdc 733
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-io 617
This theorem depends on definitions:  df-bi 110  df-dc 734
This theorem is referenced by:  bijadc  769
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