Theorem sylan9ssr 3207

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 3206	. 2 |- ( ( ph /\ ps ) -> A C_ C )
4	3	ancoms 268	1 |- ( ( ps /\ ph ) -> A C_ C )

Syntax hints:   -> wi 4   /\ wa 104   C_ wss 3166

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-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-11 1529  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-in 3172  df-ss 3179

This theorem is referenced by:  intssuni2m  3909