Theorem pm5.6r 917

Description: Conjunction in antecedent versus disjunction in consequent. One direction of Theorem *5.6 of [WhiteheadRussell] p. 125. If 𝜓 is decidable, the converse also holds (see pm5.6dc 916). (Contributed by Jim Kingdon, 4-Aug-2018.)

Assertion

Ref	Expression
pm5.6r	⊢ ((𝜑 → (𝜓 ∨ 𝜒)) → ((𝜑 ∧ ¬ 𝜓) → 𝜒))

Proof of Theorem pm5.6r

Step	Hyp	Ref	Expression
1		pm2.53 712	. . 3 ⊢ ((𝜓 ∨ 𝜒) → (¬ 𝜓 → 𝜒))
2	1	imim2i 12	. 2 ⊢ ((𝜑 → (𝜓 ∨ 𝜒)) → (𝜑 → (¬ 𝜓 → 𝜒)))
3	2	impd 252	1 ⊢ ((𝜑 → (𝜓 ∨ 𝜒)) → ((𝜑 ∧ ¬ 𝜓) → 𝜒))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ∨ wo 698

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-in2 605  ax-io 699

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  ssundifim  3492