Theorem ecase2d

Description: Deduction for elimination by cases. (Contributed by NM, 21-Apr-1994) (Proof shortened by Wolf Lammen, 22-Dec-2012)

	Ref	Expression
Hypotheses	ecase2d.1	|- ( ph -> ps )
	ecase2d.2	|- ( ph -> -. ( ps /\ ch ) )
	ecase2d.3	|- ( ph -> -. ( ps /\ th ) )
	ecase2d.4	|- ( ph -> ( ta \/ ( ch \/ th ) ) )
Assertion	ecase2d	|- ( ph -> ta )

Proof

Step	Hyp	Ref	Expression
1		ecase2d.1	|- ( ph -> ps )
2		ecase2d.2	|- ( ph -> -. ( ps /\ ch ) )
3		ecase2d.3	|- ( ph -> -. ( ps /\ th ) )
4		ecase2d.4	|- ( ph -> ( ta \/ ( ch \/ th ) ) )
5		idd	|- ( ph -> ( ta -> ta ) )
6	2	pm2.21d	|- ( ph -> ( ( ps /\ ch ) -> ta ) )
7	1 6	mpand	|- ( ph -> ( ch -> ta ) )
8	3	pm2.21d	|- ( ph -> ( ( ps /\ th ) -> ta ) )
9	1 8	mpand	|- ( ph -> ( th -> ta ) )
10	7 9	jaod	|- ( ph -> ( ( ch \/ th ) -> ta ) )
11	5 10 4	mpjaod	|- ( ph -> ta )