Theorem ecase23d 1436

Description: Deduction for elimination by cases. (Contributed by NM, 22-Apr-1994.)

Hypotheses

Ref	Expression
ecase23d.1	|- ( ph -> -. ch )
ecase23d.2	|- ( ph -> -. th )
ecase23d.3	|- ( ph -> ( ps \/ ch \/ th ) )

Assertion

Ref	Expression
ecase23d	|- ( ph -> ps )

Proof of Theorem ecase23d

Step	Hyp	Ref	Expression
1		ecase23d.1	. . 3 |- ( ph -> -. ch )
2		ecase23d.2	. . 3 |- ( ph -> -. th )
3		ioran 511	. . 3 |- ( -. ( ch \/ th ) <-> ( -. ch /\ -. th ) )
4	1, 2, 3	sylanbrc 698	. 2 |- ( ph -> -. ( ch \/ th ) )
5		ecase23d.3	. . . 4 |- ( ph -> ( ps \/ ch \/ th ) )
6		3orass 1040	. . . 4 |- ( ( ps \/ ch \/ th ) <-> ( ps \/ ( ch \/ th ) ) )
7	5, 6	sylib 208	. . 3 |- ( ph -> ( ps \/ ( ch \/ th ) ) )
8	7	ord 392	. 2 |- ( ph -> ( -. ps -> ( ch \/ th ) ) )
9	4, 8	mt3d 140	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-an 386  df-3or 1038

This theorem is referenced by:  tz7.7  5749  wfrlem10  7424  archiabllem2b  29750  nolt02o  31845  noresle  31846  nosupbnd1lem6  31859  nosupbnd2lem1  31861