Theorem anandir 599

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 400	. . 3 |- ( ( ch /\ ch ) <-> ch )
2	1	anbi2i 461	. 2 |- ( ( ( ph /\ ps ) /\ ( ch /\ ch ) ) <-> ( ( ph /\ ps ) /\ ch ) )
3		an4 592	. 2 |- ( ( ( ph /\ ps ) /\ ( ch /\ ch ) ) <-> ( ( ph /\ ch ) /\ ( ps /\ ch ) ) )
4	2, 3	bitr3i 186	1 |- ( ( ( ph /\ ps ) /\ ch ) <-> ( ( ph /\ ch ) /\ ( ps /\ ch ) ) )

This proof depends on syntax axioms:   /\ wa 104   <-> wb 105

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108

This proof depends on definitions:  df-bi 117

This theorem is used by:  anandi3r  1023  fununi  5449  imadiflem  5460  imadif  5461  imainlem  5462  elfzuzb  10422