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Theorem elssabg 4182
Description: Membership in a class abstraction involving a subset. Unlike elabg 2928,  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 4176 . . . 4  |-  ( ( A  C_  B  /\  B  e.  V )  ->  A  e.  _V )
21expcom 424 . . 3  |-  ( B  e.  V  ->  ( A  C_  B  ->  A  e.  _V ) )
32adantrd 454 . 2  |-  ( B  e.  V  ->  (
( A  C_  B  /\  ps )  ->  A  e.  _V ) )
4 sseq1 3212 . . . 4  |-  ( x  =  A  ->  (
x  C_  B  <->  A  C_  B
) )
5 elssabg.1 . . . 4  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
64, 5anbi12d 691 . . 3  |-  ( x  =  A  ->  (
( x  C_  B  /\  ph )  <->  ( A  C_  B  /\  ps )
) )
76elab3g 2933 . 2  |-  ( ( ( A  C_  B  /\  ps )  ->  A  e.  _V )  ->  ( A  e.  { x  |  ( x  C_  B  /\  ph ) }  <-> 
( A  C_  B  /\  ps ) ) )
83, 7syl 15 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 176    /\ wa 358    = wceq 1632    e. wcel 1696   {cab 2282   _Vcvv 2801    C_ wss 3165
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-v 2803  df-in 3172  df-ss 3179
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