Theorem sylan9ssr 3161

Description: A subclass transitivity deduction. (Contributed by NM, 27-Sep-2004.)

Hypotheses

Ref	Expression
sylan9ssr.1	|- ( ph -> A C_ B )
sylan9ssr.2	|- ( ps -> B C_ C )

Assertion

Ref	Expression
sylan9ssr	|- ( ( ps /\ ph ) -> A C_ C )

Proof of Theorem sylan9ssr

Step	Hyp	Ref	Expression
1		sylan9ssr.1	. . 3 |- ( ph -> A C_ B )
2		sylan9ssr.2	. . 3 |- ( ps -> B C_ C )
3	1, 2	sylan9ss 3160	. 2 |- ( ( ph /\ ps ) -> A C_ C )
4	3	ancoms 266	1 |- ( ( ps /\ ph ) -> A C_ C )

Syntax hints:   -> wi 4   /\ wa 103   C_ wss 3121

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-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-11 1499  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-in 3127  df-ss 3134

This theorem is referenced by:  intssuni2m  3855