Theorem andi 818

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 712	. . 3 |- ( ( ph /\ ps ) -> ( ( ph /\ ps ) \/ ( ph /\ ch ) ) )
2		olc 711	. . 3 |- ( ( ph /\ ch ) -> ( ( ph /\ ps ) \/ ( ph /\ ch ) ) )
3	1, 2	jaodan 797	. 2 |- ( ( ph /\ ( ps \/ ch ) ) -> ( ( ph /\ ps ) \/ ( ph /\ ch ) ) )
4		orc 712	. . . 4 |- ( ps -> ( ps \/ ch ) )
5	4	anim2i 342	. . 3 |- ( ( ph /\ ps ) -> ( ph /\ ( ps \/ ch ) ) )
6		olc 711	. . . 4 |- ( ch -> ( ps \/ ch ) )
7	6	anim2i 342	. . 3 |- ( ( ph /\ ch ) -> ( ph /\ ( ps \/ ch ) ) )
8	5, 7	jaoi 716	. 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 708

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 709

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  andir  819  anddi  821  dcim  841  dcan  933  excxor  1378  sbequilem  1838  sborv  1890  r19.43  2635  indi  3383  difindiss  3390  unrab  3407  unipr  3824  uniun  3829  unopab  4083  xpundi  4683  coundir  5132  unpreima  5642  tpostpos  6265  elni2  7313  elznn0nn  9267