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Theorem impidc 755
 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 𝜒𝜑) → (𝜓𝜒))
32con3d 561 . . . 4 ((DECID 𝜒𝜑) → (¬ 𝜒 → ¬ 𝜓))
43ex 108 . . 3 (DECID 𝜒 → (𝜑 → (¬ 𝜒 → ¬ 𝜓)))
54com23 72 . 2 (DECID 𝜒 → (¬ 𝜒 → (𝜑 → ¬ 𝜓)))
6 con1dc 753 . 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 742 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 630 This theorem depends on definitions:  df-bi 110  df-dc 743 This theorem is referenced by:  simprimdc  756
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