Theorem pm2.5dc 762

Description: Negating an implication for a decidable antecedent. Based on theorem *2.5 of [WhiteheadRussell] p. 107. (Contributed by Jim Kingdon, 29-Mar-2018.)

Assertion

Ref	Expression
pm2.5dc	⊢ (DECID φ → (¬ (φ → ψ) → (¬ φ → ψ)))

Proof of Theorem pm2.5dc

Step	Hyp	Ref	Expression
1		simplimdc 756	. . . 4 ⊢ (DECID φ → (¬ (φ → ψ) → φ))
2	1	imp 115	. . 3 ⊢ ((DECID φ ∧ ¬ (φ → ψ)) → φ)
3	2	pm2.24d 552	. 2 ⊢ ((DECID φ ∧ ¬ (φ → ψ)) → (¬ φ → ψ))
4	3	ex 108	1 ⊢ (DECID φ → (¬ (φ → ψ) → (¬ φ → ψ)))

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:  pm5.11dc  814