Theorem anandi 590

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  993  moanim  2116  difundi  3412  inrab  3432  uniin  3856  xpcom  5213  fin  5441  fndmin  5666  nnaord  6564  ixpin  6779  ltexprlemdisj  7668  bldisj  14580  blininf  14603  lgsquadlem3  15236