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Theorem ssabral 3406
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 3405 . 2  |-  ( A 
C_  { x  | 
ph }  <->  A. x
( x  e.  A  ->  ph ) )
2 df-ral 2702 . 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 1549    e. wcel 1725   {cab 2421   A.wral 2697    C_ wss 3312
This theorem is referenced by:  txdis1cn  17659  divstgplem  18142  xrhmeo  18963  cncmet  19267  itg1addlem4  19583  subfacp1lem6  24863  comppfsc  26378  istotbnd3  26471  sstotbnd  26475  heibor1lem  26509  heibor1  26510
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ral 2702  df-in 3319  df-ss 3326
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