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Theorem impidc 754
Description: An importation inference for a decidable consequent. (Contributed by Jim Kingdon, 30-Apr-2018.)
Hypothesis
Ref Expression
impidc.1 (DECID χ → (φ → (ψχ)))
Assertion
Ref Expression
impidc (DECID χ → (¬ (φ → ¬ ψ) → χ))

Proof of Theorem impidc
StepHypRef Expression
1 impidc.1 . . . . . 6 (DECID χ → (φ → (ψχ)))
21imp 115 . . . . 5 ((DECID χ φ) → (ψχ))
32con3d 560 . . . 4 ((DECID χ φ) → (¬ χ → ¬ ψ))
43ex 108 . . 3 (DECID χ → (φ → (¬ χ → ¬ ψ)))
54com23 72 . 2 (DECID χ → (¬ χ → (φ → ¬ ψ)))
6 con1dc 752 . 2 (DECID χ → ((¬ χ → (φ → ¬ ψ)) → (¬ (φ → ¬ ψ) → χ)))
75, 6mpd 13 1 (DECID χ → (¬ (φ → ¬ ψ) → χ))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wa 97  DECID wdc 741
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-in1 544  ax-in2 545  ax-io 629
This theorem depends on definitions:  df-bi 110  df-dc 742
This theorem is referenced by:  simprimdc  755
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