Theorem pm4.52im 722

Description: One direction of theorem *4.52 of [WhiteheadRussell] p. 120. The converse also holds in classical logic. (Contributed by Jim Kingdon, 27-Jul-2018.)

Assertion

Ref	Expression
pm4.52im	⊢ ((𝜑 ∧ ¬ 𝜓) → ¬ (¬ 𝜑 ∨ 𝜓))

Proof of Theorem pm4.52im

Step	Hyp	Ref	Expression
1		annimim 658	. 2 ⊢ ((𝜑 ∧ ¬ 𝜓) → ¬ (𝜑 → 𝜓))
2		imorr 693	. 2 ⊢ ((¬ 𝜑 ∨ 𝜓) → (𝜑 → 𝜓))
3	1, 2	nsyl 600	1 ⊢ ((𝜑 ∧ ¬ 𝜓) → ¬ (¬ 𝜑 ∨ 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ∨ wo 680

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  pm4.53r  723