Theorem ecase2d

Description: Deduction for elimination by cases. (Contributed by NM, 21-Apr-1994) (Proof shortened by Wolf Lammen, 19-Sep-2024)

	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	1 2	mpnanrd	|- ( ph -> -. ch )
6	1 3	mpnanrd	|- ( ph -> -. th )
7	4	ord	|- ( ph -> ( -. ta -> ( ch \/ th ) ) )
8	5 6 7	mtord	|- ( ph -> -. -. ta )
9	8	notnotrd	|- ( ph -> ta )