Metamath Proof Explorer


Theorem anandir

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

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

Proof

Step Hyp Ref Expression
1 anidm
 |-  ( ( ch /\ ch ) <-> ch )
2 1 anbi2i
 |-  ( ( ( ph /\ ps ) /\ ( ch /\ ch ) ) <-> ( ( ph /\ ps ) /\ ch ) )
3 an4
 |-  ( ( ( ph /\ ps ) /\ ( ch /\ ch ) ) <-> ( ( ph /\ ch ) /\ ( ps /\ ch ) ) )
4 2 3 bitr3i
 |-  ( ( ( ph /\ ps ) /\ ch ) <-> ( ( ph /\ ch ) /\ ( ps /\ ch ) ) )