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Theorem ssrab 3145
Description: Subclass of a restricted class abstraction. (Contributed by NM, 16-Aug-2006.)
Assertion
Ref Expression
ssrab  |-  ( B 
C_  { x  e.  A  |  ph }  <->  ( B  C_  A  /\  A. x  e.  B  ph ) )
Distinct variable groups:    x, A    x, B
Allowed substitution hint:    ph( x)

Proof of Theorem ssrab
StepHypRef Expression
1 df-rab 2402 . . 3  |-  { x  e.  A  |  ph }  =  { x  |  ( x  e.  A  /\  ph ) }
21sseq2i 3094 . 2  |-  ( B 
C_  { x  e.  A  |  ph }  <->  B 
C_  { x  |  ( x  e.  A  /\  ph ) } )
3 ssab 3137 . 2  |-  ( B 
C_  { x  |  ( x  e.  A  /\  ph ) }  <->  A. x
( x  e.  B  ->  ( x  e.  A  /\  ph ) ) )
4 dfss3 3057 . . . 4  |-  ( B 
C_  A  <->  A. x  e.  B  x  e.  A )
54anbi1i 453 . . 3  |-  ( ( B  C_  A  /\  A. x  e.  B  ph ) 
<->  ( A. x  e.  B  x  e.  A  /\  A. x  e.  B  ph ) )
6 r19.26 2535 . . 3  |-  ( A. x  e.  B  (
x  e.  A  /\  ph )  <->  ( A. x  e.  B  x  e.  A  /\  A. x  e.  B  ph ) )
7 df-ral 2398 . . 3  |-  ( A. x  e.  B  (
x  e.  A  /\  ph )  <->  A. x ( x  e.  B  ->  (
x  e.  A  /\  ph ) ) )
85, 6, 73bitr2ri 208 . 2  |-  ( A. x ( x  e.  B  ->  ( x  e.  A  /\  ph )
)  <->  ( B  C_  A  /\  A. x  e.  B  ph ) )
92, 3, 83bitri 205 1  |-  ( B 
C_  { x  e.  A  |  ph }  <->  ( B  C_  A  /\  A. x  e.  B  ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104   A.wal 1314    e. wcel 1465   {cab 2103   A.wral 2393   {crab 2397    C_ wss 3041
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rab 2402  df-in 3047  df-ss 3054
This theorem is referenced by:  ssrabdv  3146  frind  4244  epttop  12186
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