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Theorem ssab 3555
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 2583 . . 3  |-  { x  |  x  e.  A }  =  A
21sseq1i 3513 . 2  |-  ( { x  |  x  e.  A }  C_  { x  |  ph }  <->  A  C_  { x  |  ph } )
3 ss2ab 3553 . 2  |-  ( { x  |  x  e.  A }  C_  { x  |  ph }  <->  A. x
( x  e.  A  ->  ph ) )
42, 3bitr3i 251 1  |-  ( A 
C_  { x  | 
ph }  <->  A. x
( x  e.  A  ->  ph ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184   A.wal 1381    e. wcel 1804   {cab 2428    C_ wss 3461
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-in 3468  df-ss 3475
This theorem is referenced by:  ssabral  3556  ssrab  3563  wdomd  8010  ixpiunwdom  8020  lidldvgen  17882  prdsxmslem2  21010  ballotlem2  28405
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