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Theorem ssrab 3097
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 2368 . . 3  |-  { x  e.  A  |  ph }  =  { x  |  ( x  e.  A  /\  ph ) }
21sseq2i 3049 . 2  |-  ( B 
C_  { x  e.  A  |  ph }  <->  B 
C_  { x  |  ( x  e.  A  /\  ph ) } )
3 ssab 3089 . 2  |-  ( B 
C_  { x  |  ( x  e.  A  /\  ph ) }  <->  A. x
( x  e.  B  ->  ( x  e.  A  /\  ph ) ) )
4 dfss3 3013 . . . 4  |-  ( B 
C_  A  <->  A. x  e.  B  x  e.  A )
54anbi1i 446 . . 3  |-  ( ( B  C_  A  /\  A. x  e.  B  ph ) 
<->  ( A. x  e.  B  x  e.  A  /\  A. x  e.  B  ph ) )
6 r19.26 2497 . . 3  |-  ( A. x  e.  B  (
x  e.  A  /\  ph )  <->  ( A. x  e.  B  x  e.  A  /\  A. x  e.  B  ph ) )
7 df-ral 2364 . . 3  |-  ( A. x  e.  B  (
x  e.  A  /\  ph )  <->  A. x ( x  e.  B  ->  (
x  e.  A  /\  ph ) ) )
85, 6, 73bitr2ri 207 . 2  |-  ( A. x ( x  e.  B  ->  ( x  e.  A  /\  ph )
)  <->  ( B  C_  A  /\  A. x  e.  B  ph ) )
92, 3, 83bitri 204 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 102    <-> wb 103   A.wal 1287    e. wcel 1438   {cab 2074   A.wral 2359   {crab 2363    C_ wss 2997
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rab 2368  df-in 3003  df-ss 3010
This theorem is referenced by:  ssrabdv  3098  frind  4170
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