Theorem ecase3ad 986

Description: Deduction for elimination by cases. (Contributed by NM, 24-May-2013.)

Hypotheses

Ref	Expression
ecase3ad.1	|- ( ph -> ( ps -> th ) )
ecase3ad.2	|- ( ph -> ( ch -> th ) )
ecase3ad.3	|- ( ph -> ( ( -. ps /\ -. ch ) -> th ) )

Assertion

Ref	Expression
ecase3ad	|- ( ph -> th )

Proof of Theorem ecase3ad

Step	Hyp	Ref	Expression
1		notnotr 125	. . 3 |- ( -. -. ps -> ps )
2		ecase3ad.1	. . 3 |- ( ph -> ( ps -> th ) )
3	1, 2	syl5 34	. 2 |- ( ph -> ( -. -. ps -> th ) )
4		notnotr 125	. . 3 |- ( -. -. ch -> ch )
5		ecase3ad.2	. . 3 |- ( ph -> ( ch -> th ) )
6	4, 5	syl5 34	. 2 |- ( ph -> ( -. -. ch -> th ) )
7		ecase3ad.3	. 2 |- ( ph -> ( ( -. ps /\ -. ch ) -> th ) )
8	3, 6, 7	ecased 985	1 |- ( ph -> th )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384

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

This theorem is referenced by: (None)