Theorem ordir 835

Description: Distributive law for disjunction. (Contributed by NM, 12-Aug-1994.)

Assertion

Ref	Expression
ordir	⊢ (((φ ∧ ψ) ∨ χ) ↔ ((φ ∨ χ) ∧ (ψ ∨ χ)))

Proof of Theorem ordir

Step	Hyp	Ref	Expression
1		ordi 834	. 2 ⊢ ((χ ∨ (φ ∧ ψ)) ↔ ((χ ∨ φ) ∧ (χ ∨ ψ)))
2		orcom 376	. 2 ⊢ (((φ ∧ ψ) ∨ χ) ↔ (χ ∨ (φ ∧ ψ)))
3		orcom 376	. . 3 ⊢ ((φ ∨ χ) ↔ (χ ∨ φ))
4		orcom 376	. . 3 ⊢ ((ψ ∨ χ) ↔ (χ ∨ ψ))
5	3, 4	anbi12i 678	. 2 ⊢ (((φ ∨ χ) ∧ (ψ ∨ χ)) ↔ ((χ ∨ φ) ∧ (χ ∨ ψ)))
6	1, 2, 5	3bitr4i 268	1 ⊢ (((φ ∧ ψ) ∨ χ) ↔ ((φ ∨ χ) ∧ (ψ ∨ χ)))

Syntax hints:   ↔ 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:  orddi  839  pm5.62  889  dn1  932  cadan  1392  nchoicelem9  6298