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Theorem jadc 748
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 747 . . 3 (DECID φ → ((¬ φχ) → ((φχ) → χ)))
53, 4mpd 13 . 2 (DECID φ → ((φχ) → χ))
62, 5syl5 28 1 (DECID φ → ((φψ) → χ))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  DECID wdc 730
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 731
This theorem is referenced by:  pm5.71dc  854
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