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  1389  sbequilem  1849  sborv  1902  r19.43  2648  indi  3397  difindiss  3404  unrab  3421  unipr  3838  uniun  3843  unopab  4097  xpundi  4700  coundir  5149  unpreima  5662  tpostpos  6289  elni2  7343  elznn0nn  9297