Theorem andir 838

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

Assertion

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

Proof of Theorem andir

Step	Hyp	Ref	Expression
1		andi 837	. 2 ⊢ ((χ ∧ (φ ∨ ψ)) ↔ ((χ ∧ φ) ∨ (χ ∧ ψ)))
2		ancom 437	. 2 ⊢ (((φ ∨ ψ) ∧ χ) ↔ (χ ∧ (φ ∨ ψ)))
3		ancom 437	. . 3 ⊢ ((φ ∧ χ) ↔ (χ ∧ φ))
4		ancom 437	. . 3 ⊢ ((ψ ∧ χ) ↔ (χ ∧ ψ))
5	3, 4	orbi12i 507	. 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:  anddi  840  cador  1391  rexun  3444  rabun2  3535  reuun2  3539  elimif  3692  xpundir  4834  coundi  5083  nchoicelem18  6307