Theorem andir 809

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 808	. 2 |- ( ( ch /\ ( ph \/ ps ) ) <-> ( ( ch /\ ph ) \/ ( ch /\ ps ) ) )
2		ancom 264	. 2 |- ( ( ( ph \/ ps ) /\ ch ) <-> ( ch /\ ( ph \/ ps ) ) )
3		ancom 264	. . 3 |- ( ( ph /\ ch ) <-> ( ch /\ ph ) )
4		ancom 264	. . 3 |- ( ( ps /\ ch ) <-> ( ch /\ ps ) )
5	3, 4	orbi12i 754	. 2 |- ( ( ( ph /\ ch ) \/ ( ps /\ ch ) ) <-> ( ( ch /\ ph ) \/ ( ch /\ ps ) ) )
6	1, 2, 5	3bitr4i 211	1 |- ( ( ( ph \/ ps ) /\ ch ) <-> ( ( ph /\ ch ) \/ ( ps /\ ch ) ) )

Syntax hints:   /\ wa 103   <-> wb 104   \/ wo 698

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  anddi  811  dcan  923  excxor  1368  xordc1  1383  sbequilem  1826  rexun  3301  rabun2  3400  reuun2  3404  xpundir  4660  coundi  5104  mptun  5318  tpostpos  6228  ltxr  9707  pythagtriplem2  12194  pythagtrip  12211