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Theorem ssab 3065
Description: Subclass of a class abstraction. (Contributed by NM, 16-Aug-2006.)
Assertion
Ref Expression
ssab  |-  ( A 
C_  { x  | 
ph }  <->  A. x
( x  e.  A  ->  ph ) )
Distinct variable group:    x, A
Allowed substitution hint:    ph( x)

Proof of Theorem ssab
StepHypRef Expression
1 abid2 2200 . . 3  |-  { x  |  x  e.  A }  =  A
21sseq1i 3024 . 2  |-  ( { x  |  x  e.  A }  C_  { x  |  ph }  <->  A  C_  { x  |  ph } )
3 ss2ab 3063 . 2  |-  ( { x  |  x  e.  A }  C_  { x  |  ph }  <->  A. x
( x  e.  A  ->  ph ) )
42, 3bitr3i 184 1  |-  ( A 
C_  { x  | 
ph }  <->  A. x
( x  e.  A  ->  ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103   A.wal 1283    e. wcel 1434   {cab 2068    C_ wss 2974
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-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-in 2980  df-ss 2987
This theorem is referenced by:  ssabral  3066  ssrab  3073
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