Theorem andir

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

	Ref	Expression
Assertion	andir	|- ( ( ( ph \/ ps ) /\ ch ) <-> ( ( ph /\ ch ) \/ ( ps /\ ch ) ) )

Proof

Step	Hyp	Ref	Expression
1		andi	|- ( ( ch /\ ( ph \/ ps ) ) <-> ( ( ch /\ ph ) \/ ( ch /\ ps ) ) )
2		ancom	|- ( ( ( ph \/ ps ) /\ ch ) <-> ( ch /\ ( ph \/ ps ) ) )
3		ancom	|- ( ( ph /\ ch ) <-> ( ch /\ ph ) )
4		ancom	|- ( ( ps /\ ch ) <-> ( ch /\ ps ) )
5	3 4	orbi12i	|- ( ( ( ph /\ ch ) \/ ( ps /\ ch ) ) <-> ( ( ch /\ ph ) \/ ( ch /\ ps ) ) )
6	1 2 5	3bitr4i	|- ( ( ( ph \/ ps ) /\ ch ) <-> ( ( ph /\ ch ) \/ ( ps /\ ch ) ) )