Theorem ccased 913

Description: Deduction for combining cases. (Contributed by NM, 9-May-2004.)

Hypotheses

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

Assertion

Ref	Expression
ccased	|- (ph -> (((ps \/ th) /\ (ch \/ ta)) -> et))

Proof of Theorem ccased

Step	Hyp	Ref	Expression
1		ccased.1	. . . 4 |- (ph -> ((ps /\ ch) -> et))
2	1	com12 27	. . 3 |- ((ps /\ ch) -> (ph -> et))
3		ccased.2	. . . 4 |- (ph -> ((th /\ ch) -> et))
4	3	com12 27	. . 3 |- ((th /\ ch) -> (ph -> et))
5		ccased.3	. . . 4 |- (ph -> ((ps /\ ta) -> et))
6	5	com12 27	. . 3 |- ((ps /\ ta) -> (ph -> et))
7		ccased.4	. . . 4 |- (ph -> ((th /\ ta) -> et))
8	7	com12 27	. . 3 |- ((th /\ ta) -> (ph -> et))
9	2, 4, 6, 8	ccase 912	. 2 |- (((ps \/ th) /\ (ch \/ ta)) -> (ph -> et))
10	9	com12 27	1 |- (ph -> (((ps \/ th) /\ (ch \/ ta)) -> et))

Syntax hints:   -> wi 4   \/ wo 357   /\ wa 358

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 177  df-or 359  df-an 360

This theorem is referenced by: (None)