Theorem pm5.6 878

Description: Conjunction in antecedent versus disjunction in consequent. Theorem *5.6 of [WhiteheadRussell] p. 125. (Contributed by NM, 8-Jun-1994.)

Assertion

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

Proof of Theorem pm5.6

Step	Hyp	Ref	Expression
1		impexp 433	. 2 ⊢ (((φ ∧ ¬ ψ) → χ) ↔ (φ → (¬ ψ → χ)))
2		df-or 359	. . 3 ⊢ ((ψ ∨ χ) ↔ (¬ ψ → χ))
3	2	imbi2i 303	. 2 ⊢ ((φ → (ψ ∨ χ)) ↔ (φ → (¬ ψ → χ)))
4	1, 3	bitr4i 243	1 ⊢ (((φ ∧ ¬ ψ) → χ) ↔ (φ → (ψ ∨ χ)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 176   ∨ wo 357   ∧ wa 358

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 177  df-or 359  df-an 360

This theorem is used by:  ssundif  3634