Theorem 3anbi123i

Description: Join 3 biconditionals with conjunction. (Contributed by NM, 21-Apr-1994)

	Ref	Expression
Hypotheses	bi3.1	|- ( ph <-> ps )
	bi3.2	|- ( ch <-> th )
	bi3.3	|- ( ta <-> et )
Assertion	3anbi123i	|- ( ( ph /\ ch /\ ta ) <-> ( ps /\ th /\ et ) )

Proof

Step	Hyp	Ref	Expression
1		bi3.1	|- ( ph <-> ps )
2		bi3.2	|- ( ch <-> th )
3		bi3.3	|- ( ta <-> et )
4	1 2	anbi12i	|- ( ( ph /\ ch ) <-> ( ps /\ th ) )
5	4 3	anbi12i	|- ( ( ( ph /\ ch ) /\ ta ) <-> ( ( ps /\ th ) /\ et ) )
6		df-3an	|- ( ( ph /\ ch /\ ta ) <-> ( ( ph /\ ch ) /\ ta ) )
7		df-3an	|- ( ( ps /\ th /\ et ) <-> ( ( ps /\ th ) /\ et ) )
8	5 6 7	3bitr4i	|- ( ( ph /\ ch /\ ta ) <-> ( ps /\ th /\ et ) )