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Theorem elssabg 3925
Description: Membership in a class abstraction involving a subset. Unlike elabg 2740,  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 3919 . . . 4  |-  ( ( A  C_  B  /\  B  e.  V )  ->  A  e.  _V )
21expcom 114 . . 3  |-  ( B  e.  V  ->  ( A  C_  B  ->  A  e.  _V ) )
32adantrd 273 . 2  |-  ( B  e.  V  ->  (
( A  C_  B  /\  ps )  ->  A  e.  _V ) )
4 sseq1 3021 . . . 4  |-  ( x  =  A  ->  (
x  C_  B  <->  A  C_  B
) )
5 elssabg.1 . . . 4  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
64, 5anbi12d 457 . . 3  |-  ( x  =  A  ->  (
( x  C_  B  /\  ph )  <->  ( A  C_  B  /\  ps )
) )
76elab3g 2745 . 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 102    <-> wb 103    = wceq 1285    e. wcel 1434   {cab 2068   _Vcvv 2602    C_ wss 2974
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3898
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-in 2980  df-ss 2987
This theorem is referenced by: (None)
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