Theorem andir 827

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 826	. 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 772	. 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 716

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 717

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  anddi  829  excxor  1423  xordc1  1438  sbequilem  1887  rexun  3403  rabun2  3504  reuun2  3508  xpundir  4812  coundi  5269  mptun  5495  tpostpos  6508  ltxr  10127  hashfibclem  11231  pythagtriplem2  12989  pythagtrip  13006  vtxdfifiun  16418