Theorem ecase2d 906

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

Hypotheses

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

Assertion

Ref	Expression
ecase2d	|- (ph -> ta)

Proof of Theorem ecase2d

Step	Hyp	Ref	Expression
1		idd 21	. 2 |- (ph -> (ta -> ta))
2		ecase2d.1	. . . 4 |- (ph -> ps)
3		ecase2d.2	. . . . 5 |- (ph -> -. (ps /\ ch))
4	3	pm2.21d 98	. . . 4 |- (ph -> ((ps /\ ch) -> ta))
5	2, 4	mpand 656	. . 3 |- (ph -> (ch -> ta))
6		ecase2d.3	. . . . 5 |- (ph -> -. (ps /\ th))
7	6	pm2.21d 98	. . . 4 |- (ph -> ((ps /\ th) -> ta))
8	2, 7	mpand 656	. . 3 |- (ph -> (th -> ta))
9	5, 8	jaod 369	. 2 |- (ph -> ((ch \/ th) -> ta))
10		ecase2d.4	. 2 |- (ph -> (ta \/ (ch \/ th)))
11	1, 9, 10	mpjaod 370	1 |- (ph -> ta)

Syntax hints:  -. wn 3   -> wi 4   \/ wo 357   /\ wa 358

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360

This theorem is referenced by: (None)