Theorem ccase2 564

Description: Inference for combining cases.

Hypotheses

Ref	Expression
ccase2.1	|- ((ph /\ ps) -> ta )
ccase2.2	|- (ch -> ta )
ccase2.3	|- (th -> ta )

Assertion

Ref	Expression
ccase2	|- (((ph \/ ch) /\ (ps \/ th)) -> ta )

Proof of Theorem ccase2

Step	Hyp	Ref	Expression
1		ccase2.1	. 2 |- ((ph /\ ps) -> ta )
2		ccase2.2	. . 3 |- (ch -> ta )
3	2	adantr 306	. 2 |- ((ch /\ ps) -> ta )
4		ccase2.3	. . 3 |- (th -> ta )
5	4	adantl 305	. 2 |- ((ph /\ th) -> ta )
6	4	adantl 305	. 2 |- ((ch /\ th) -> ta )
7	1, 3, 5, 6	ccase 562	1 |- (((ph \/ ch) /\ (ps \/ th)) -> ta )

Syntax hints:   -> wi 2   \/ wo 195   /\ wa 196

This theorem is referenced by:  add20 4329  mulge0 4335  nn0mulcl 4553

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198

metamath.org