Theorem andir 765

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 764	. 2 |- ( ( ch /\ ( ph \/ ps ) ) <-> ( ( ch /\ ph ) \/ ( ch /\ ps ) ) )
2		ancom 262	. 2 |- ( ( ( ph \/ ps ) /\ ch ) <-> ( ch /\ ( ph \/ ps ) ) )
3		ancom 262	. . 3 |- ( ( ph /\ ch ) <-> ( ch /\ ph ) )
4		ancom 262	. . 3 |- ( ( ps /\ ch ) <-> ( ch /\ ps ) )
5	3, 4	orbi12i 713	. 2 |- ( ( ( ph /\ ch ) \/ ( ps /\ ch ) ) <-> ( ( ch /\ ph ) \/ ( ch /\ ps ) ) )
6	1, 2, 5	3bitr4i 210	1 |- ( ( ( ph \/ ps ) /\ ch ) <-> ( ( ph /\ ch ) \/ ( ps /\ ch ) ) )

Syntax hints:   /\ wa 102   <-> wb 103   \/ wo 661

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662

This theorem depends on definitions:  df-bi 115

This theorem is referenced by:  anddi  767  dcan  875  excxor  1309  xordc1  1324  sbequilem  1759  rexun  3152  rabun2  3243  reuun2  3247  xpundir  4415  coundi  4842  mptun  5049  tpostpos  5902  ltxr  8849