Theorem pm4.78 565

Description: Theorem *4.78 of [WhiteheadRussell] p. 121. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 19-Nov-2012.)

Assertion

Ref	Expression
pm4.78	⊢ (((φ → ψ) ∨ (φ → χ)) ↔ (φ → (ψ ∨ χ)))

Proof of Theorem pm4.78

Step	Hyp	Ref	Expression
1		orordi 516	. 2 ⊢ ((¬ φ ∨ (ψ ∨ χ)) ↔ ((¬ φ ∨ ψ) ∨ (¬ φ ∨ χ)))
2		imor 401	. 2 ⊢ ((φ → (ψ ∨ χ)) ↔ (¬ φ ∨ (ψ ∨ χ)))
3		imor 401	. . 3 ⊢ ((φ → ψ) ↔ (¬ φ ∨ ψ))
4		imor 401	. . 3 ⊢ ((φ → χ) ↔ (¬ φ ∨ χ))
5	3, 4	orbi12i 507	. 2 ⊢ (((φ → ψ) ∨ (φ → χ)) ↔ ((¬ φ ∨ ψ) ∨ (¬ φ ∨ χ)))
6	1, 2, 5	3bitr4ri 269	1 ⊢ (((φ → ψ) ∨ (φ → χ)) ↔ (φ → (ψ ∨ χ)))

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

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

This theorem is referenced by: (None)