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  2652  indi  3407  difindiss  3414  unrab  3431  unipr  3850  uniun  3855  unopab  4109  xpundi  4716  coundir  5169  unpreima  5684  tpostpos  6319  elni2  7376  elznn0nn  9334  lgsquadlem3  15236