Theorem andi 768

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 669	. . 3 |- ( ( ph /\ ps ) -> ( ( ph /\ ps ) \/ ( ph /\ ch ) ) )
2		olc 668	. . 3 |- ( ( ph /\ ch ) -> ( ( ph /\ ps ) \/ ( ph /\ ch ) ) )
3	1, 2	jaodan 747	. 2 |- ( ( ph /\ ( ps \/ ch ) ) -> ( ( ph /\ ps ) \/ ( ph /\ ch ) ) )
4		orc 669	. . . 4 |- ( ps -> ( ps \/ ch ) )
5	4	anim2i 335	. . 3 |- ( ( ph /\ ps ) -> ( ph /\ ( ps \/ ch ) ) )
6		olc 668	. . . 4 |- ( ch -> ( ps \/ ch ) )
7	6	anim2i 335	. . 3 |- ( ( ph /\ ch ) -> ( ph /\ ( ps \/ ch ) ) )
8	5, 7	jaoi 672	. 2 |- ( ( ( ph /\ ps ) \/ ( ph /\ ch ) ) -> ( ph /\ ( ps \/ ch ) ) )
9	3, 8	impbii 125	1 |- ( ( ph /\ ( ps \/ ch ) ) <-> ( ( ph /\ ps ) \/ ( ph /\ ch ) ) )

Syntax hints:   /\ wa 103   <-> wb 104   \/ wo 665

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  andir  769  anddi  771  dcim  823  dcan  881  excxor  1315  sbequilem  1767  sborv  1819  r19.43  2526  indi  3247  difindiss  3254  unrab  3271  unipr  3673  uniun  3678  unopab  3923  xpundi  4507  coundir  4946  unpreima  5438  tpostpos  6043  elni2  6934  elznn0nn  8825