Theorem 3jcad

Description: Deduction conjoining the consequents of three implications. (Contributed by NM, 25-Sep-2005)

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

Proof

Step	Hyp	Ref	Expression
1		3jcad.1	|- ( ph -> ( ps -> ch ) )
2		3jcad.2	|- ( ph -> ( ps -> th ) )
3		3jcad.3	|- ( ph -> ( ps -> ta ) )
4	1	imp	|- ( ( ph /\ ps ) -> ch )
5	2	imp	|- ( ( ph /\ ps ) -> th )
6	3	imp	|- ( ( ph /\ ps ) -> ta )
7	4 5 6	3jca	|- ( ( ph /\ ps ) -> ( ch /\ th /\ ta ) )
8	7	ex	|- ( ph -> ( ps -> ( ch /\ th /\ ta ) ) )