Metamath Proof Explorer


Theorem 3anbi12d

Description: Deduction conjoining and adding a conjunct to equivalences. (Contributed by NM, 8-Sep-2006)

Ref Expression
Hypotheses 3anbi12d.1
|- ( ph -> ( ps <-> ch ) )
3anbi12d.2
|- ( ph -> ( th <-> ta ) )
Assertion 3anbi12d
|- ( ph -> ( ( ps /\ th /\ et ) <-> ( ch /\ ta /\ et ) ) )

Proof

Step Hyp Ref Expression
1 3anbi12d.1
 |-  ( ph -> ( ps <-> ch ) )
2 3anbi12d.2
 |-  ( ph -> ( th <-> ta ) )
3 biidd
 |-  ( ph -> ( et <-> et ) )
4 1 2 3 3anbi123d
 |-  ( ph -> ( ( ps /\ th /\ et ) <-> ( ch /\ ta /\ et ) ) )