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Theorem sylan9ssr 3039
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 3038 . 2  |-  ( (
ph  /\  ps )  ->  A  C_  C )
43ancoms 264 1  |-  ( ( ps  /\  ph )  ->  A  C_  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    C_ wss 2999
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 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-11 1442  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-in 3005  df-ss 3012
This theorem is referenced by:  intssuni2m  3710
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