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Theorem elssabg 4355
Description: Membership in a class abstraction involving a subset. Unlike elabg 3083,  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 4349 . . . 4  |-  ( ( A  C_  B  /\  B  e.  V )  ->  A  e.  _V )
21expcom 425 . . 3  |-  ( B  e.  V  ->  ( A  C_  B  ->  A  e.  _V ) )
32adantrd 455 . 2  |-  ( B  e.  V  ->  (
( A  C_  B  /\  ps )  ->  A  e.  _V ) )
4 sseq1 3369 . . . 4  |-  ( x  =  A  ->  (
x  C_  B  <->  A  C_  B
) )
5 elssabg.1 . . . 4  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
64, 5anbi12d 692 . . 3  |-  ( x  =  A  ->  (
( x  C_  B  /\  ph )  <->  ( A  C_  B  /\  ps )
) )
76elab3g 3088 . 2  |-  ( ( ( A  C_  B  /\  ps )  ->  A  e.  _V )  ->  ( A  e.  { x  |  ( x  C_  B  /\  ph ) }  <-> 
( A  C_  B  /\  ps ) ) )
83, 7syl 16 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    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   {cab 2422   _Vcvv 2956    C_ wss 3320
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-v 2958  df-in 3327  df-ss 3334
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