Theorem anandir 586

Description: Distribution of conjunction over conjunction. (Contributed by NM, 24-Aug-1995.)

Assertion

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

Proof of Theorem anandir

Step	Hyp	Ref	Expression
1		anidm 394	. . 3 |- ( ( ch /\ ch ) <-> ch )
2	1	anbi2i 454	. 2 |- ( ( ( ph /\ ps ) /\ ( ch /\ ch ) ) <-> ( ( ph /\ ps ) /\ ch ) )
3		an4 581	. 2 |- ( ( ( ph /\ ps ) /\ ( ch /\ ch ) ) <-> ( ( ph /\ ch ) /\ ( ps /\ ch ) ) )
4	2, 3	bitr3i 185	1 |- ( ( ( ph /\ ps ) /\ ch ) <-> ( ( ph /\ ch ) /\ ( ps /\ 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:  anandi3r  987  fununi  5266  imadiflem  5277  imadif  5278  imainlem  5279  elfzuzb  9975