Theorem anandi 592

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 396	. . 3 |- ( ( ph /\ ph ) <-> ph )
2	1	anbi1i 458	. 2 |- ( ( ( ph /\ ph ) /\ ( ps /\ ch ) ) <-> ( ph /\ ( ps /\ ch ) ) )
3		an4 586	. 2 |- ( ( ( ph /\ ph ) /\ ( ps /\ ch ) ) <-> ( ( ph /\ ps ) /\ ( ph /\ ch ) ) )
4	2, 3	bitr3i 186	1 |- ( ( ph /\ ( ps /\ ch ) ) <-> ( ( ph /\ ps ) /\ ( ph /\ ch ) ) )

Syntax hints:   /\ wa 104   <-> wb 105

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  anandi3  1015  moanim  2152  difundi  3456  inrab  3476  uniin  3908  xpcom  5275  fin  5512  fndmin  5742  nnaord  6655  ixpin  6870  ltexprlemdisj  7793  bldisj  15075  blininf  15098  lgsquadlem3  15758  wlkeq  16065