Theorem anandir 555

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 388	. . 3 |- ( ( ch /\ ch ) <-> ch )
2	1	anbi2i 444	. 2 |- ( ( ( ph /\ ps ) /\ ( ch /\ ch ) ) <-> ( ( ph /\ ps ) /\ ch ) )
3		an4 550	. 2 |- ( ( ( ph /\ ps ) /\ ( ch /\ ch ) ) <-> ( ( ph /\ ch ) /\ ( ps /\ ch ) ) )
4	2, 3	bitr3i 184	1 |- ( ( ( ph /\ ps ) /\ ch ) <-> ( ( ph /\ ch ) /\ ( ps /\ ch ) ) )

Syntax hints:   /\ wa 102   <-> wb 103

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106

This theorem depends on definitions:  df-bi 115

This theorem is referenced by:  anandi3r  933  fununi  4987  imadiflem  4998  imadif  4999  imainlem  5000  elfzuzb  9039