Theorem pm4.79 566

Description: Theorem *4.79 of [WhiteheadRussell] p. 121. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 27-Jun-2013.)

Assertion

Ref	Expression
pm4.79	⊢ (((ψ → φ) ∨ (χ → φ)) ↔ ((ψ ∧ χ) → φ))

Proof of Theorem pm4.79

Step	Hyp	Ref	Expression
1		id 19	. . 3 ⊢ ((ψ → φ) → (ψ → φ))
2		id 19	. . 3 ⊢ ((χ → φ) → (χ → φ))
3	1, 2	jaoa 496	. 2 ⊢ (((ψ → φ) ∨ (χ → φ)) → ((ψ ∧ χ) → φ))
4		simplim 143	. . . 4 ⊢ (¬ (ψ → φ) → ψ)
5		pm3.3 431	. . . 4 ⊢ (((ψ ∧ χ) → φ) → (ψ → (χ → φ)))
6	4, 5	syl5 28	. . 3 ⊢ (((ψ ∧ χ) → φ) → (¬ (ψ → φ) → (χ → φ)))
7	6	orrd 367	. 2 ⊢ (((ψ ∧ χ) → φ) → ((ψ → φ) ∨ (χ → φ)))
8	3, 7	impbii 180	1 ⊢ (((ψ → φ) ∨ (χ → φ)) ↔ ((ψ ∧ χ) → φ))

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)