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Theorem pm2.5gdc 852
 Description: Negating an implication for a decidable antecedent. General instance of Theorem *2.5 of [WhiteheadRussell] p. 107 under a decidability condition. (Contributed by Jim Kingdon, 29-Mar-2018.)
Assertion
Ref Expression
pm2.5gdc (DECID 𝜑 → (¬ (𝜑𝜓) → (¬ 𝜑𝜒)))

Proof of Theorem pm2.5gdc
StepHypRef Expression
1 simplimdc 846 . . . 4 (DECID 𝜑 → (¬ (𝜑𝜓) → 𝜑))
21imp 123 . . 3 ((DECID 𝜑 ∧ ¬ (𝜑𝜓)) → 𝜑)
32pm2.24d 612 . 2 ((DECID 𝜑 ∧ ¬ (𝜑𝜓)) → (¬ 𝜑𝜒))
43ex 114 1 (DECID 𝜑 → (¬ (𝜑𝜓) → (¬ 𝜑𝜒)))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103  DECID wdc 820 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699 This theorem depends on definitions:  df-bi 116  df-stab 817  df-dc 821 This theorem is referenced by:  pm2.5dc  853  pm2.521gdc  854
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