Theorem pm5.71 902

Description: Theorem *5.71 of [WhiteheadRussell] p. 125. (Contributed by Roy F. Longton, 23-Jun-2005.)

Assertion

Ref	Expression
pm5.71	⊢ ((ψ → ¬ χ) → (((φ ∨ ψ) ∧ χ) ↔ (φ ∧ χ)))

Proof of Theorem pm5.71

Step	Hyp	Ref	Expression
1		orel2 372	. . . 4 ⊢ (¬ ψ → ((φ ∨ ψ) → φ))
2		orc 374	. . . 4 ⊢ (φ → (φ ∨ ψ))
3	1, 2	impbid1 194	. . 3 ⊢ (¬ ψ → ((φ ∨ ψ) ↔ φ))
4	3	anbi1d 685	. 2 ⊢ (¬ ψ → (((φ ∨ ψ) ∧ χ) ↔ (φ ∧ χ)))
5		pm2.21 100	. . 3 ⊢ (¬ χ → (χ → ((φ ∨ ψ) ↔ φ)))
6	5	pm5.32rd 621	. 2 ⊢ (¬ χ → (((φ ∨ ψ) ∧ χ) ↔ (φ ∧ χ)))
7	4, 6	ja 153	1 ⊢ ((ψ → ¬ χ) → (((φ ∨ ψ) ∧ χ) ↔ (φ ∧ χ)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∨ wo 357   ∧ wa 358

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

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

This theorem is referenced by: (None)