Theorem ecase13d 15350

Description: Deduction for elimination by cases.

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 861	. . . . . 6 |- ((ch \/ ps \/ th) <-> (ch \/ (ps \/ th)))
5		df-or 241	. . . . . 6 |- ((ch \/ (ps \/ th)) <-> (-. ch -> (ps \/ th)))
6	4, 5	bitri 190	. . . . 5 |- ((ch \/ ps \/ th) <-> (-. ch -> (ps \/ th)))
7	3, 6	sylib 215	. . . 4 |- (ph -> (-. ch -> (ps \/ th)))
8	2, 7	mpd 29	. . 3 |- (ph -> (ps \/ th))
9		orcom 266	. . . 4 |- ((ps \/ th) <-> (th \/ ps))
10		df-or 241	. . . 4 |- ((th \/ ps) <-> (-. th -> ps))
11	9, 10	bitri 190	. . 3 |- ((ps \/ th) <-> (-. th -> ps))
12	8, 11	sylib 215	. 2 |- (ph -> (-. th -> ps))
13	1, 12	mpd 29	1 |- (ph -> ps)

Syntax hints:  -. wn 2   -> wi 3   \/ wo 239   \/ w3o 857

This theorem is referenced by:  ivthALT 15454

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7

This theorem depends on definitions:  df-bi 164  df-or 241  df-3or 859

Copyright terms: Public domain