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Theorem jadc 771
Description: Inference forming an implication from the antecedents of two premises, where a decidable antecedent is negated. (Contributed by Jim Kingdon, 25-Mar-2018.)
Hypotheses
Ref Expression
jadc.1 (DECID 𝜑 → (¬ 𝜑𝜒))
jadc.2 (𝜓𝜒)
Assertion
Ref Expression
jadc (DECID 𝜑 → ((𝜑𝜓) → 𝜒))

Proof of Theorem jadc
StepHypRef Expression
1 jadc.2 . . 3 (𝜓𝜒)
21imim2i 12 . 2 ((𝜑𝜓) → (𝜑𝜒))
3 jadc.1 . . 3 (DECID 𝜑 → (¬ 𝜑𝜒))
4 pm2.6dc 770 . . 3 (DECID 𝜑 → ((¬ 𝜑𝜒) → ((𝜑𝜒) → 𝜒)))
53, 4mpd 13 . 2 (DECID 𝜑 → ((𝜑𝜒) → 𝜒))
62, 5syl5 32 1 (DECID 𝜑 → ((𝜑𝜓) → 𝜒))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  DECID wdc 753
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640
This theorem depends on definitions:  df-bi 114  df-dc 754
This theorem is referenced by:  pm5.71dc  879
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