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Theorem elssabg 4043
Description: Membership in a class abstraction involving a subset. Unlike elabg 2803,  A does not have to be a set. (Contributed by NM, 29-Aug-2006.)
Hypothesis
Ref Expression
elssabg.1  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
elssabg  |-  ( B  e.  V  ->  ( A  e.  { x  |  ( x  C_  B  /\  ph ) }  <-> 
( A  C_  B  /\  ps ) ) )
Distinct variable groups:    x, A    x, B    ps, x
Allowed substitution hints:    ph( x)    V( x)

Proof of Theorem elssabg
StepHypRef Expression
1 ssexg 4037 . . . 4  |-  ( ( A  C_  B  /\  B  e.  V )  ->  A  e.  _V )
21expcom 115 . . 3  |-  ( B  e.  V  ->  ( A  C_  B  ->  A  e.  _V ) )
32adantrd 277 . 2  |-  ( B  e.  V  ->  (
( A  C_  B  /\  ps )  ->  A  e.  _V ) )
4 sseq1 3090 . . . 4  |-  ( x  =  A  ->  (
x  C_  B  <->  A  C_  B
) )
5 elssabg.1 . . . 4  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
64, 5anbi12d 464 . . 3  |-  ( x  =  A  ->  (
( x  C_  B  /\  ph )  <->  ( A  C_  B  /\  ps )
) )
76elab3g 2808 . 2  |-  ( ( ( A  C_  B  /\  ps )  ->  A  e.  _V )  ->  ( A  e.  { x  |  ( x  C_  B  /\  ph ) }  <-> 
( A  C_  B  /\  ps ) ) )
83, 7syl 14 1  |-  ( B  e.  V  ->  ( A  e.  { x  |  ( x  C_  B  /\  ph ) }  <-> 
( A  C_  B  /\  ps ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1316    e. wcel 1465   {cab 2103   _Vcvv 2660    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  ax-sep 4016
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-in 3047  df-ss 3054
This theorem is referenced by: (None)
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