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Theorem jaddc 799
Description: Deduction forming an implication from the antecedents of two premises, where a decidable antecedent is negated. (Contributed by Jim Kingdon, 26-Mar-2018.)
Hypotheses
Ref Expression
jaddc.1 (𝜑 → (DECID 𝜓 → (¬ 𝜓𝜃)))
jaddc.2 (𝜑 → (𝜒𝜃))
Assertion
Ref Expression
jaddc (𝜑 → (DECID 𝜓 → ((𝜓𝜒) → 𝜃)))

Proof of Theorem jaddc
StepHypRef Expression
1 jaddc.2 . . 3 (𝜑 → (𝜒𝜃))
21imim2d 53 . 2 (𝜑 → ((𝜓𝜒) → (𝜓𝜃)))
3 jaddc.1 . . 3 (𝜑 → (DECID 𝜓 → (¬ 𝜓𝜃)))
4 pm2.6dc 797 . . 3 (DECID 𝜓 → ((¬ 𝜓𝜃) → ((𝜓𝜃) → 𝜃)))
53, 4sylcom 28 . 2 (𝜑 → (DECID 𝜓 → ((𝜓𝜃) → 𝜃)))
62, 5syl5d 67 1 (𝜑 → (DECID 𝜓 → ((𝜓𝜒) → 𝜃)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  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-io 665
This theorem depends on definitions:  df-bi 115  df-dc 781
This theorem is referenced by:  pm2.54dc  828
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