Theorem andir 820

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 819	. 2 |- ( ( ch /\ ( ph \/ ps ) ) <-> ( ( ch /\ ph ) \/ ( ch /\ ps ) ) )
2		ancom 266	. 2 |- ( ( ( ph \/ ps ) /\ ch ) <-> ( ch /\ ( ph \/ ps ) ) )
3		ancom 266	. . 3 |- ( ( ph /\ ch ) <-> ( ch /\ ph ) )
4		ancom 266	. . 3 |- ( ( ps /\ ch ) <-> ( ch /\ ps ) )
5	3, 4	orbi12i 765	. 2 |- ( ( ( ph /\ ch ) \/ ( ps /\ ch ) ) <-> ( ( ch /\ ph ) \/ ( ch /\ ps ) ) )
6	1, 2, 5	3bitr4i 212	1 |- ( ( ( ph \/ ps ) /\ ch ) <-> ( ( ph /\ ch ) \/ ( ps /\ ch ) ) )

Syntax hints:   /\ wa 104   <-> wb 105   \/ wo 709

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  anddi  822  excxor  1389  xordc1  1404  sbequilem  1849  rexun  3339  rabun2  3438  reuun2  3442  xpundir  4716  coundi  5167  mptun  5385  tpostpos  6317  ltxr  9841  pythagtriplem2  12404  pythagtrip  12421