Theorem pm5.6r 836

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 835). (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		pm2.53 641	. . 3 ⊢ ((𝜓 ∨ 𝜒) → (¬ 𝜓 → 𝜒))
2	1	imim2i 12	. 2 ⊢ ((𝜑 → (𝜓 ∨ 𝜒)) → (𝜑 → (¬ 𝜓 → 𝜒)))
3	2	impd 242	1 ⊢ ((𝜑 → (𝜓 ∨ 𝜒)) → ((𝜑 ∧ ¬ 𝜓) → 𝜒))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   ∨ wo 629

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-in2 545  ax-io 630

This theorem depends on definitions:  df-bi 110

This theorem is referenced by:  ssundifim  3306