Theorem ecase23d 1391

Description: Variation of ecased 1390 with three disjuncts instead of two. (Contributed by NM, 22-Apr-1994.) (Revised by Jim Kingdon, 9-Dec-2017.)

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	. 2 |- ( ph -> -. ch )
2		ecase23d.2	. . 3 |- ( ph -> -. th )
3		ecase23d.3	. . . 4 |- ( ph -> ( ps \/ ch \/ th ) )
4		df-3or 1010	. . . 4 |- ( ( ps \/ ch \/ th ) <-> ( ( ps \/ ch ) \/ th ) )
5	3, 4	sylib 122	. . 3 |- ( ph -> ( ( ps \/ ch ) \/ th ) )
6	2, 5	ecased 1390	. 2 |- ( ph -> ( ps \/ ch ) )
7	1, 6	ecased 1390	1 |- ( ph -> ps )

This proof depends on syntax axioms:   -. wn 3   -> wi 4   \/ wo 720   \/ w3o 1008

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in2 624  ax-io 721

This proof depends on definitions:  df-bi 117  df-3or 1010

This theorem is used by:  iseqf1olemklt  10948  xrmaxiflemcl  12027  xrmaxifle  12028  xrmaxiflemab  12029  xrmaxiflemlub  12030  ennnfonelemex  13354  mulgval  13974  mulgfng  13976  subgmulg  14040