Theorem ecase23d 1285

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 476	. . 3 |- (-. (ch \/ th) <-> (-. ch /\ -. th))
4	1, 2, 3	sylanbrc 645	. 2 |- (ph -> -. (ch \/ th))
5		ecase23d.3	. . . 4 |- (ph -> (ps \/ ch \/ th))
6		3orass 937	. . . 4 |- ((ps \/ ch \/ th) <-> (ps \/ (ch \/ th)))
7	5, 6	sylib 188	. . 3 |- (ph -> (ps \/ (ch \/ th)))
8	7	ord 366	. 2 |- (ph -> (-. ps -> (ch \/ th)))
9	4, 8	mt3d 117	1 |- (ph -> ps)

Syntax hints:  -. wn 3   -> wi 4   \/ wo 357   \/ w3o 933

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  df-3or 935

This theorem is referenced by:  tfinltfin  4502