Metamath Proof Explorer


Theorem anandi

Description: Distribution of conjunction over conjunction. (Contributed by NM, 14-Aug-1995)

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

Proof

Step Hyp Ref Expression
1 anidm
 |-  ( ( ph /\ ph ) <-> ph )
2 1 anbi1i
 |-  ( ( ( ph /\ ph ) /\ ( ps /\ ch ) ) <-> ( ph /\ ( ps /\ ch ) ) )
3 an4
 |-  ( ( ( ph /\ ph ) /\ ( ps /\ ch ) ) <-> ( ( ph /\ ps ) /\ ( ph /\ ch ) ) )
4 2 3 bitr3i
 |-  ( ( ph /\ ( ps /\ ch ) ) <-> ( ( ph /\ ps ) /\ ( ph /\ ch ) ) )