Theorem ccase 948

Description: Inference for combining cases. (Contributed by NM, 29-Jul-1999.) (Proof shortened by Wolf Lammen, 6-Jan-2013.)

Hypotheses

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

Assertion

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

Proof of Theorem ccase

Step	Hyp	Ref	Expression
1		ccase.1	. . 3 |- ( ( ph /\ ps ) -> ta )
2		ccase.2	. . 3 |- ( ( ch /\ ps ) -> ta )
3	1, 2	jaoian 784	. 2 |- ( ( ( ph \/ ch ) /\ ps ) -> ta )
4		ccase.3	. . 3 |- ( ( ph /\ th ) -> ta )
5		ccase.4	. . 3 |- ( ( ch /\ th ) -> ta )
6	4, 5	jaoian 784	. 2 |- ( ( ( ph \/ ch ) /\ th ) -> ta )
7	3, 6	jaodan 786	1 |- ( ( ( ph \/ ch ) /\ ( ps \/ th ) ) -> ta )

Syntax hints:   -> wi 4   /\ wa 103   \/ wo 697

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  ccased  949  ccase2  950  undif3ss  3337  prodmodc  11347  nn0gcdsq  11878