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Theorem ssabral 3576
Description: The relation for a subclass of a class abstraction is equivalent to restricted quantification. (Contributed by NM, 6-Sep-2006.)
Assertion
Ref Expression
ssabral  |-  ( A 
C_  { x  | 
ph }  <->  A. x  e.  A  ph )
Distinct variable group:    x, A
Allowed substitution hint:    ph( x)

Proof of Theorem ssabral
StepHypRef Expression
1 ssab 3575 . 2  |-  ( A 
C_  { x  | 
ph }  <->  A. x
( x  e.  A  ->  ph ) )
2 df-ral 2822 . 2  |-  ( A. x  e.  A  ph  <->  A. x
( x  e.  A  ->  ph ) )
31, 2bitr4i 252 1  |-  ( A 
C_  { x  | 
ph }  <->  A. x  e.  A  ph )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184   A.wal 1377    e. wcel 1767   {cab 2452   A.wral 2817    C_ wss 3481
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2822  df-in 3488  df-ss 3495
This theorem is referenced by:  comppfsc  19878  txdis1cn  19981  qustgplem  20464  xrhmeo  21291  cncmet  21606  itg1addlem4  21951  subfacp1lem6  28422  istotbnd3  30162  sstotbnd  30166  heibor1lem  30200  heibor1  30201
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