Theorem 3ecase 1437

Description: Inference for elimination by cases. (Contributed by NM, 13-Jul-2005.)

Hypotheses

Ref	Expression
3ecase.1	|- ( -. ph -> th )
3ecase.2	|- ( -. ps -> th )
3ecase.3	|- ( -. ch -> th )
3ecase.4	|- ( ( ph /\ ps /\ ch ) -> th )

Assertion

Ref	Expression
3ecase	|- th

Proof of Theorem 3ecase

Step	Hyp	Ref	Expression
1		3ecase.4	. . . 4 |- ( ( ph /\ ps /\ ch ) -> th )
2	1	3exp 1264	. . 3 |- ( ph -> ( ps -> ( ch -> th ) ) )
3		3ecase.1	. . . 4 |- ( -. ph -> th )
4	3	2a1d 26	. . 3 |- ( -. ph -> ( ps -> ( ch -> th ) ) )
5	2, 4	pm2.61i 176	. 2 |- ( ps -> ( ch -> th ) )
6		3ecase.2	. 2 |- ( -. ps -> th )
7		3ecase.3	. 2 |- ( -. ch -> th )
8	5, 6, 7	pm2.61nii 178	1 |- th

Syntax hints:   -. wn 3   -> wi 4   /\ w3a 1037

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-an 386  df-3an 1039

This theorem is referenced by: (None)