Theorem 3anim123d

Description: Deduction joining 3 implications to form implication of conjunctions. (Contributed by NM, 24-Feb-2005)

	Ref	Expression
Hypotheses	3anim123d.1	|- ( ph -> ( ps -> ch ) )
	3anim123d.2	|- ( ph -> ( th -> ta ) )
	3anim123d.3	|- ( ph -> ( et -> ze ) )
Assertion	3anim123d	|- ( ph -> ( ( ps /\ th /\ et ) -> ( ch /\ ta /\ ze ) ) )

Proof

Step	Hyp	Ref	Expression
1		3anim123d.1	|- ( ph -> ( ps -> ch ) )
2		3anim123d.2	|- ( ph -> ( th -> ta ) )
3		3anim123d.3	|- ( ph -> ( et -> ze ) )
4	1 2	anim12d	|- ( ph -> ( ( ps /\ th ) -> ( ch /\ ta ) ) )
5	4 3	anim12d	|- ( ph -> ( ( ( ps /\ th ) /\ et ) -> ( ( ch /\ ta ) /\ ze ) ) )
6		df-3an	|- ( ( ps /\ th /\ et ) <-> ( ( ps /\ th ) /\ et ) )
7		df-3an	|- ( ( ch /\ ta /\ ze ) <-> ( ( ch /\ ta ) /\ ze ) )
8	5 6 7	3imtr4g	|- ( ph -> ( ( ps /\ th /\ et ) -> ( ch /\ ta /\ ze ) ) )