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Theorem sylan9ssr 3193
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
StepHypRef Expression
1 sylan9ssr.1 . . 3  |-  ( ph  ->  A  C_  B )
2 sylan9ssr.2 . . 3  |-  ( ps 
->  B  C_  C )
31, 2sylan9ss 3192 . 2  |-  ( (
ph  /\  ps )  ->  A  C_  C )
43ancoms 439 1  |-  ( ( ps  /\  ph )  ->  A  C_  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    C_ wss 3152
This theorem is referenced by:  intssuni2  3887  marypha1  7187  cardinfima  7724  cfflb  7885  ssfin4  7936  acsfn  13561  mrelatlub  14289  efgval  15026  islbs3  15908  kgentopon  17233  txlly  17330  topjoin  26314  filnetlem3  26329  sspwimpALT  28701  bnj1014  28992
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-in 3159  df-ss 3166
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