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Theorem elab3 2934
Description: Membership in a class abstraction using implicit substitution. (Contributed by NM, 10-Nov-2000.)
Hypotheses
Ref Expression
elab3.1  |-  ( ps 
->  A  e.  _V )
elab3.2  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
elab3  |-  ( A  e.  { x  | 
ph }  <->  ps )
Distinct variable groups:    ps, x    x, A
Allowed substitution hint:    ph( x)

Proof of Theorem elab3
StepHypRef Expression
1 elab3.1 . 2  |-  ( ps 
->  A  e.  _V )
2 elab3.2 . . 3  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
32elab3g 2933 . 2  |-  ( ( ps  ->  A  e.  _V )  ->  ( A  e.  { x  | 
ph }  <->  ps )
)
41, 3ax-mp 8 1  |-  ( A  e.  { x  | 
ph }  <->  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1632    e. wcel 1696   {cab 2282   _Vcvv 2801
This theorem is referenced by:  fvelrnb  5586  elrnmpt2  5973  ovelrn  6012  isfi  6901  isnum2  7594  pm54.43lem  7648  isfin3  7938  isfin5  7941  isfin6  7942  genpelv  8640  iswrd  11431  4sqlem2  13012  vdwapval  13036  ismnd  14385  isghm  14699  issrng  15631  lspsnel  15776  lspprel  15863  iscss  16599  istps  16690  islp  16888  is2ndc  17188  elpt  17283  itg2l  19100  elply  19593  vtarsuelt  25998  ellspd  27357  rngunsnply  27481  isline  30550  ispointN  30553  ispsubsp  30556  ispsubclN  30748  islaut  30894  ispautN  30910  istendo  31571
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
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