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Theorem ssab 3350
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 2498 . . 3  |-  { x  |  x  e.  A }  =  A
21sseq1i 3309 . 2  |-  ( { x  |  x  e.  A }  C_  { x  |  ph }  <->  A  C_  { x  |  ph } )
3 ss2ab 3348 . 2  |-  ( { x  |  x  e.  A }  C_  { x  |  ph }  <->  A. x
( x  e.  A  ->  ph ) )
42, 3bitr3i 243 1  |-  ( A 
C_  { x  | 
ph }  <->  A. x
( x  e.  A  ->  ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177   A.wal 1546    e. wcel 1717   {cab 2367    C_ wss 3257
This theorem is referenced by:  ssabral  3351  ssrab  3358  wdomd  7476  ixpiunwdom  7486  lidldvgen  16247  prdsxmslem2  18443  ballotlem2  24519
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 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2362
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2506  df-in 3264  df-ss 3271
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