Theorem ecase13d 32307

Description: Deduction for elimination by cases. (Contributed by Jeff Hankins, 18-Aug-2009.)

Hypotheses

Ref	Expression
ecase13d.1	|- ( ph -> -. ch )
ecase13d.2	|- ( ph -> -. th )
ecase13d.3	|- ( ph -> ( ch \/ ps \/ th ) )

Assertion

Ref	Expression
ecase13d	|- ( ph -> ps )

Proof of Theorem ecase13d

Step	Hyp	Ref	Expression
1		ecase13d.2	. 2 |- ( ph -> -. th )
2		ecase13d.1	. . . 4 |- ( ph -> -. ch )
3		ecase13d.3	. . . . 5 |- ( ph -> ( ch \/ ps \/ th ) )
4		3orass 1040	. . . . . 6 |- ( ( ch \/ ps \/ th ) <-> ( ch \/ ( ps \/ th ) ) )
5		df-or 385	. . . . . 6 |- ( ( ch \/ ( ps \/ th ) ) <-> ( -. ch -> ( ps \/ th ) ) )
6	4, 5	bitri 264	. . . . 5 |- ( ( ch \/ ps \/ th ) <-> ( -. ch -> ( ps \/ th ) ) )
7	3, 6	sylib 208	. . . 4 |- ( ph -> ( -. ch -> ( ps \/ th ) ) )
8	2, 7	mpd 15	. . 3 |- ( ph -> ( ps \/ th ) )
9		orcom 402	. . . 4 |- ( ( ps \/ th ) <-> ( th \/ ps ) )
10		df-or 385	. . . 4 |- ( ( th \/ ps ) <-> ( -. th -> ps ) )
11	9, 10	bitri 264	. . 3 |- ( ( ps \/ th ) <-> ( -. th -> ps ) )
12	8, 11	sylib 208	. 2 |- ( ph -> ( -. th -> ps ) )
13	1, 12	mpd 15	1 |- ( ph -> ps )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 383   \/ w3o 1036

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 197  df-or 385  df-3or 1038

This theorem is referenced by:  ivthALT  32330