Theorem andir 732

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

Assertion

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

Proof of Theorem andir

Step	Hyp	Ref	Expression
1		andi 731	. 2 |- ( ( ch /\ ( ph \/ ps ) ) <-> ( ( ch /\ ph ) \/ ( ch /\ ps ) ) )
2		ancom 253	. 2 |- ( ( ( ph \/ ps ) /\ ch ) <-> ( ch /\ ( ph \/ ps ) ) )
3		ancom 253	. . 3 |- ( ( ph /\ ch ) <-> ( ch /\ ph ) )
4		ancom 253	. . 3 |- ( ( ps /\ ch ) <-> ( ch /\ ps ) )
5	3, 4	orbi12i 681	. 2 |- ( ( ( ph /\ ch ) \/ ( ps /\ ch ) ) <-> ( ( ch /\ ph ) \/ ( ch /\ ps ) ) )
6	1, 2, 5	3bitr4i 201	1 |- ( ( ( ph \/ ps ) /\ ch ) <-> ( ( ph /\ ch ) \/ ( ps /\ ch ) ) )

Syntax hints:   /\ wa 97   <-> wb 98   \/ wo 629

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630

This theorem depends on definitions:  df-bi 110

This theorem is referenced by:  anddi  734  dcan  842  excxor  1269  xordc1  1284  sbequilem  1719  rexun  3123  rabun2  3216  reuun2  3220  xpundir  4397  coundi  4822  mptun  5029  tpostpos  5879  ltxr  8695