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Theorem ssabral 3241
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 3240 . 2  |-  ( A 
C_  { x  | 
ph }  <->  A. x
( x  e.  A  ->  ph ) )
2 df-ral 2473 . 2  |-  ( A. x  e.  A  ph  <->  A. x
( x  e.  A  ->  ph ) )
31, 2bitr4i 187 1  |-  ( A 
C_  { x  | 
ph }  <->  A. x  e.  A  ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 105   A.wal 1362    e. wcel 2160   {cab 2175   A.wral 2468    C_ wss 3144
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-in 3150  df-ss 3157
This theorem is referenced by:  txdis1cn  14215  bj-bdfindis  15136
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