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Theorem elabgt 2924
Description: Membership in a class abstraction, using implicit substitution. (Closed theorem version of elabg 2928.) (Contributed by NM, 7-Nov-2005.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
Assertion
Ref Expression
elabgt  |-  ( ( A  e.  B  /\  A. x ( x  =  A  ->  ( ph  <->  ps ) ) )  -> 
( A  e.  {
x  |  ph }  <->  ps ) )
Distinct variable groups:    x, A    ps, x
Allowed substitution hints:    ph( x)    B( x)

Proof of Theorem elabgt
StepHypRef Expression
1 abid 2284 . . . . . . 7  |-  ( x  e.  { x  | 
ph }  <->  ph )
2 eleq1 2356 . . . . . . 7  |-  ( x  =  A  ->  (
x  e.  { x  |  ph }  <->  A  e.  { x  |  ph }
) )
31, 2syl5bbr 250 . . . . . 6  |-  ( x  =  A  ->  ( ph 
<->  A  e.  { x  |  ph } ) )
43bibi1d 310 . . . . 5  |-  ( x  =  A  ->  (
( ph  <->  ps )  <->  ( A  e.  { x  |  ph } 
<->  ps ) ) )
54biimpd 198 . . . 4  |-  ( x  =  A  ->  (
( ph  <->  ps )  ->  ( A  e.  { x  |  ph }  <->  ps )
) )
65a2i 12 . . 3  |-  ( ( x  =  A  -> 
( ph  <->  ps ) )  -> 
( x  =  A  ->  ( A  e. 
{ x  |  ph } 
<->  ps ) ) )
76alimi 1549 . 2  |-  ( A. x ( x  =  A  ->  ( ph  <->  ps ) )  ->  A. x
( x  =  A  ->  ( A  e. 
{ x  |  ph } 
<->  ps ) ) )
8 nfcv 2432 . . . 4  |-  F/_ x A
9 nfab1 2434 . . . . . 6  |-  F/_ x { x  |  ph }
109nfel2 2444 . . . . 5  |-  F/ x  A  e.  { x  |  ph }
11 nfv 1609 . . . . 5  |-  F/ x ps
1210, 11nfbi 1784 . . . 4  |-  F/ x
( A  e.  {
x  |  ph }  <->  ps )
13 pm5.5 326 . . . 4  |-  ( x  =  A  ->  (
( x  =  A  ->  ( A  e. 
{ x  |  ph } 
<->  ps ) )  <->  ( A  e.  { x  |  ph } 
<->  ps ) ) )
148, 12, 13spcgf 2876 . . 3  |-  ( A  e.  B  ->  ( A. x ( x  =  A  ->  ( A  e.  { x  |  ph } 
<->  ps ) )  -> 
( A  e.  {
x  |  ph }  <->  ps ) ) )
1514imp 418 . 2  |-  ( ( A  e.  B  /\  A. x ( x  =  A  ->  ( A  e.  { x  |  ph } 
<->  ps ) ) )  ->  ( A  e. 
{ x  |  ph } 
<->  ps ) )
167, 15sylan2 460 1  |-  ( ( A  e.  B  /\  A. x ( x  =  A  ->  ( ph  <->  ps ) ) )  -> 
( A  e.  {
x  |  ph }  <->  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   A.wal 1530    = wceq 1632    e. wcel 1696   {cab 2282
This theorem is referenced by:  abfmpeld  23233  abfmpel  23234  esumc  23445  dfrtrcl2  24060  eqintg  25064
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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