Theorem sylan9ssr 3013

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 3012	. 2 |- ( ( ph /\ ps ) -> A C_ C )
4	3	ancoms 264	1 |- ( ( ps /\ ph ) -> A C_ C )

Syntax hints:   -> wi 4   /\ wa 102   C_ wss 2973

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-11 1437  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-in 2979  df-ss 2986

This theorem is referenced by:  intssuni2m  3660