Theorem ccased 967

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 30	. . 3 |- ( ( ps /\ ch ) -> ( ph -> et ) )
3		ccased.2	. . . 4 |- ( ph -> ( ( th /\ ch ) -> et ) )
4	3	com12 30	. . 3 |- ( ( th /\ ch ) -> ( ph -> et ) )
5		ccased.3	. . . 4 |- ( ph -> ( ( ps /\ ta ) -> et ) )
6	5	com12 30	. . 3 |- ( ( ps /\ ta ) -> ( ph -> et ) )
7		ccased.4	. . . 4 |- ( ph -> ( ( th /\ ta ) -> et ) )
8	7	com12 30	. . 3 |- ( ( th /\ ta ) -> ( ph -> et ) )
9	2, 4, 6, 8	ccase 966	. 2 |- ( ( ( ps \/ th ) /\ ( ch \/ ta ) ) -> ( ph -> et ) )
10	9	com12 30	1 |- ( ph -> ( ( ( ps \/ th ) /\ ( ch \/ ta ) ) -> et ) )

Syntax hints:   -> wi 4   /\ wa 104   \/ wo 709

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  zmulcl  9379  gcdabs  12155  lcmabs  12244  mulgass  13289  lgsdir2lem5  15273