Theorem pm2.5gdc 851

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 → (¬ (𝜑 → 𝜓) → (¬ 𝜑 → 𝜒)))

Proof of Theorem pm2.5gdc

Step	Hyp	Ref	Expression
1		simplimdc 845	. . . 4 ⊢ (DECID 𝜑 → (¬ (𝜑 → 𝜓) → 𝜑))
2	1	imp 123	. . 3 ⊢ ((DECID 𝜑 ∧ ¬ (𝜑 → 𝜓)) → 𝜑)
3	2	pm2.24d 611	. 2 ⊢ ((DECID 𝜑 ∧ ¬ (𝜑 → 𝜓)) → (¬ 𝜑 → 𝜒))
4	3	ex 114	1 ⊢ (DECID 𝜑 → (¬ (𝜑 → 𝜓) → (¬ 𝜑 → 𝜒)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103  DECID wdc 819

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 603  ax-in2 604  ax-io 698

This theorem depends on definitions:  df-bi 116  df-stab 816  df-dc 820

This theorem is referenced by:  pm2.5dc  852  pm2.521gdc  853