Theorem andi 819

Description: Distributive law for conjunction. Theorem *4.4 of [WhiteheadRussell] p. 118. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 5-Jan-2013.)

Assertion

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

Proof of Theorem andi

Step	Hyp	Ref	Expression
1		orc 713	. . 3 |- ( ( ph /\ ps ) -> ( ( ph /\ ps ) \/ ( ph /\ ch ) ) )
2		olc 712	. . 3 |- ( ( ph /\ ch ) -> ( ( ph /\ ps ) \/ ( ph /\ ch ) ) )
3	1, 2	jaodan 798	. 2 |- ( ( ph /\ ( ps \/ ch ) ) -> ( ( ph /\ ps ) \/ ( ph /\ ch ) ) )
4		orc 713	. . . 4 |- ( ps -> ( ps \/ ch ) )
5	4	anim2i 342	. . 3 |- ( ( ph /\ ps ) -> ( ph /\ ( ps \/ ch ) ) )
6		olc 712	. . . 4 |- ( ch -> ( ps \/ ch ) )
7	6	anim2i 342	. . 3 |- ( ( ph /\ ch ) -> ( ph /\ ( ps \/ ch ) ) )
8	5, 7	jaoi 717	. 2 |- ( ( ( ph /\ ps ) \/ ( ph /\ ch ) ) -> ( ph /\ ( ps \/ ch ) ) )
9	3, 8	impbii 126	1 |- ( ( ph /\ ( ps \/ ch ) ) <-> ( ( ph /\ ps ) \/ ( ph /\ 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:  andir  820  anddi  822  dcim  842  excxor  1397  sbequilem  1860  sborv  1913  r19.43  2663  indi  3419  difindiss  3426  unrab  3443  unipr  3863  uniun  3868  unopab  4122  xpundi  4730  coundir  5184  unpreima  5704  tpostpos  6349  elni2  7426  elznn0nn  9385  lgsquadlem3  15527