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Theorem impidc 766
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 119 . . . . 5 ((DECID 𝜒𝜑) → (𝜓𝜒))
32con3d 571 . . . 4 ((DECID 𝜒𝜑) → (¬ 𝜒 → ¬ 𝜓))
43ex 112 . . 3 (DECID 𝜒 → (𝜑 → (¬ 𝜒 → ¬ 𝜓)))
54com23 76 . 2 (DECID 𝜒 → (¬ 𝜒 → (𝜑 → ¬ 𝜓)))
6 con1dc 764 . 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 101  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-in1 554  ax-in2 555  ax-io 640
This theorem depends on definitions:  df-bi 114  df-dc 754
This theorem is referenced by:  simprimdc  767
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