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Theorem elab4g 3221
Description: Membership in a class abstraction, using implicit substitution. (Contributed by NM, 17-Oct-2012.)
Hypotheses
Ref Expression
elab4g.1  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
elab4g.2  |-  B  =  { x  |  ph }
Assertion
Ref Expression
elab4g  |-  ( A  e.  B  <->  ( A  e.  _V  /\  ps )
)
Distinct variable groups:    ps, x    x, A
Allowed substitution hints:    ph( x)    B( x)

Proof of Theorem elab4g
StepHypRef Expression
1 elex 3089 . 2  |-  ( A  e.  B  ->  A  e.  _V )
2 elab4g.1 . . 3  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
3 elab4g.2 . . 3  |-  B  =  { x  |  ph }
42, 3elab2g 3219 . 2  |-  ( A  e.  _V  ->  ( A  e.  B  <->  ps )
)
51, 4biadan2 646 1  |-  ( A  e.  B  <->  ( A  e.  _V  /\  ps )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1872   {cab 2407   _Vcvv 3080
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401
This theorem depends on definitions:  df-bi 188  df-an 372  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-v 3082
This theorem is referenced by:  isprs  16175  ispos  16192  istrkgc  24501  istrkgb  24502  istrkgcb  24503  istrkge  24504  istrkgl  24505  eulerpartlemt0  29211  istrkg2d  29492
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