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Theorem ssabral 3374
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 3373 . 2  |-  ( A 
C_  { x  | 
ph }  <->  A. x
( x  e.  A  ->  ph ) )
2 df-ral 2671 . 2  |-  ( A. x  e.  A  ph  <->  A. x
( x  e.  A  ->  ph ) )
31, 2bitr4i 244 1  |-  ( A 
C_  { x  | 
ph }  <->  A. x  e.  A  ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177   A.wal 1546    e. wcel 1721   {cab 2390   A.wral 2666    C_ wss 3280
This theorem is referenced by:  txdis1cn  17620  divstgplem  18103  xrhmeo  18924  cncmet  19228  itg1addlem4  19544  subfacp1lem6  24824  comppfsc  26277  istotbnd3  26370  sstotbnd  26374  heibor1lem  26408  heibor1  26409
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 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ral 2671  df-in 3287  df-ss 3294
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