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Theorem ssab 3256
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 2413 . . 3  |-  { x  |  x  e.  A }  =  A
21sseq1i 3215 . 2  |-  ( { x  |  x  e.  A }  C_  { x  |  ph }  <->  A  C_  { x  |  ph } )
3 ss2ab 3254 . 2  |-  ( { x  |  x  e.  A }  C_  { x  |  ph }  <->  A. x
( x  e.  A  ->  ph ) )
42, 3bitr3i 242 1  |-  ( A 
C_  { x  | 
ph }  <->  A. x
( x  e.  A  ->  ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176   A.wal 1530    e. wcel 1696   {cab 2282    C_ wss 3165
This theorem is referenced by:  ssabral  3257  ssrab  3264  wdomd  7311  ixpiunwdom  7321  lidldvgen  16023  prdsxmslem2  18091  ballotlem2  23063  imgfldref2  25167  qusp  25645  pgapspf  26155
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-in 3172  df-ss 3179
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