Theorem 3ecase 919

Description: Inference for elimination by cases.

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 829	. . 3 |- (ph -> (ps -> (ch -> th)))
3		3ecase.1	. . . . 5 |- (-. ph -> th)
4	3	a1d 12	. . . 4 |- (-. ph -> (ch -> th))
5	4	a1d 12	. . 3 |- (-. ph -> (ps -> (ch -> th)))
6	2, 5	pm2.61i 126	. 2 |- (ps -> (ch -> th))
7		3ecase.2	. 2 |- (-. ps -> th)
8		3ecase.3	. 2 |- (-. ch -> th)
9	6, 7, 8	pm2.61nii 131	1 |- th

Syntax hints:  -. wn 2   -> wi 3   /\ w3a 772

This theorem is referenced by:  bcpasc 6858

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 147  df-an 225  df-3an 774

Copyright terms: Public domain