Theorem orimdi 820

Description: Disjunction distributes over implication. (Contributed by Wolf Lammen, 5-Jan-2013.)

Assertion

Ref	Expression
orimdi	⊢ ((φ ∨ (ψ → χ)) ↔ ((φ ∨ ψ) → (φ ∨ χ)))

Proof of Theorem orimdi

Step	Hyp	Ref	Expression
1		imdi 352	. 2 ⊢ ((¬ φ → (ψ → χ)) ↔ ((¬ φ → ψ) → (¬ φ → χ)))
2		df-or 359	. 2 ⊢ ((φ ∨ (ψ → χ)) ↔ (¬ φ → (ψ → χ)))
3		df-or 359	. . 3 ⊢ ((φ ∨ ψ) ↔ (¬ φ → ψ))
4		df-or 359	. . 3 ⊢ ((φ ∨ χ) ↔ (¬ φ → χ))
5	3, 4	imbi12i 316	. 2 ⊢ (((φ ∨ ψ) → (φ ∨ χ)) ↔ ((¬ φ → ψ) → (¬ φ → χ)))
6	1, 2, 5	3bitr4i 268	1 ⊢ ((φ ∨ (ψ → χ)) ↔ ((φ ∨ ψ) → (φ ∨ 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:  pm2.76  821  pm2.85  826