MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  elab4g Unicode version

Theorem elab4g 2931
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 2809 . 2  |-  ( A  e.  B  ->  A  e.  _V )
2 elab4g.1 . . 3  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
3 elab4g.2 . . 3  |-  B  =  { x  |  ph }
42, 3elab2g 2929 . 2  |-  ( A  e.  _V  ->  ( A  e.  B  <->  ps )
)
51, 4biadan2 623 1  |-  ( A  e.  B  <->  ( A  e.  _V  /\  ps )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   {cab 2282   _Vcvv 2801
This theorem is referenced by:  isprs  14080  ispos  14097  bisig0  26165
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
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
  Copyright terms: Public domain W3C validator