Theorem orddi 839

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

Assertion

Ref	Expression
orddi	⊢ (((φ ∧ ψ) ∨ (χ ∧ θ)) ↔ (((φ ∨ χ) ∧ (φ ∨ θ)) ∧ ((ψ ∨ χ) ∧ (ψ ∨ θ))))

Proof of Theorem orddi

Step	Hyp	Ref	Expression
1		ordir 835	. 2 ⊢ (((φ ∧ ψ) ∨ (χ ∧ θ)) ↔ ((φ ∨ (χ ∧ θ)) ∧ (ψ ∨ (χ ∧ θ))))
2		ordi 834	. . 3 ⊢ ((φ ∨ (χ ∧ θ)) ↔ ((φ ∨ χ) ∧ (φ ∨ θ)))
3		ordi 834	. . 3 ⊢ ((ψ ∨ (χ ∧ θ)) ↔ ((ψ ∨ χ) ∧ (ψ ∨ θ)))
4	2, 3	anbi12i 678	. 2 ⊢ (((φ ∨ (χ ∧ θ)) ∧ (ψ ∨ (χ ∧ θ))) ↔ (((φ ∨ χ) ∧ (φ ∨ θ)) ∧ ((ψ ∨ χ) ∧ (ψ ∨ θ))))
5	1, 4	bitri 240	1 ⊢ (((φ ∧ ψ) ∨ (χ ∧ θ)) ↔ (((φ ∨ χ) ∧ (φ ∨ θ)) ∧ ((ψ ∨ χ) ∧ (ψ ∨ θ))))

This proof depends on syntax axioms:   ↔ wb 176   ∨ wo 357   ∧ wa 358

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

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

This theorem is used by: (None)