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Theorem sylan9ssr 3326
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 3325 . 2  |-  ( (
ph  /\  ps )  ->  A  C_  C )
43ancoms 440 1  |-  ( ( ps  /\  ph )  ->  A  C_  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    C_ wss 3284
This theorem is referenced by:  intssuni2  4039  marypha1  7401  cardinfima  7938  cfflb  8099  ssfin4  8150  acsfn  13843  mrelatlub  14571  efgval  15308  islbs3  16186  kgentopon  17527  txlly  17625  sigaclci  24472  mblfinlem2  26148  topjoin  26288  filnetlem3  26303  sspwimpALT  28750  sspwimpALT2  28754  bnj1014  29041
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2395  df-cleq 2401  df-clel 2404  df-in 3291  df-ss 3298
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