Theorem anandi 580

Description: Distribution of conjunction over conjunction. (Contributed by NM, 14-Aug-1995.)

Assertion

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

Proof of Theorem anandi

Step	Hyp	Ref	Expression
1		anidm 394	. . 3 |- ( ( ph /\ ph ) <-> ph )
2	1	anbi1i 454	. 2 |- ( ( ( ph /\ ph ) /\ ( ps /\ ch ) ) <-> ( ph /\ ( ps /\ ch ) ) )
3		an4 576	. 2 |- ( ( ( ph /\ ph ) /\ ( ps /\ ch ) ) <-> ( ( ph /\ ps ) /\ ( ph /\ ch ) ) )
4	2, 3	bitr3i 185	1 |- ( ( ph /\ ( ps /\ ch ) ) <-> ( ( ph /\ ps ) /\ ( ph /\ ch ) ) )

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

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

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  anandi3  976  moanim  2074  difundi  3333  inrab  3353  uniin  3764  xpcom  5093  fin  5317  fndmin  5535  nnaord  6413  ixpin  6625  ltexprlemdisj  7438  bldisj  12609  blininf  12632