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Theorem ssabral 3420
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 3419 . 2  |-  ( A 
C_  { x  | 
ph }  <->  A. x
( x  e.  A  ->  ph ) )
2 df-ral 2718 . 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 1362    e. wcel 1761   {cab 2427   A.wral 2713    C_ wss 3325
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-ex 1592  df-nf 1595  df-sb 1706  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ral 2718  df-in 3332  df-ss 3339
This theorem is referenced by:  txdis1cn  19108  divstgplem  19591  xrhmeo  20418  cncmet  20733  itg1addlem4  21077  subfacp1lem6  26987  comppfsc  28488  istotbnd3  28579  sstotbnd  28583  heibor1lem  28617  heibor1  28618
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