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Theorem elabgt 3071
Description: Membership in a class abstraction, using implicit substitution. (Closed theorem version of elabg 3075.) (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 2423 . . . . . . 7  |-  ( x  e.  { x  | 
ph }  <->  ph )
2 eleq1 2495 . . . . . . 7  |-  ( x  =  A  ->  (
x  e.  { x  |  ph }  <->  A  e.  { x  |  ph }
) )
31, 2syl5bbr 251 . . . . . 6  |-  ( x  =  A  ->  ( ph 
<->  A  e.  { x  |  ph } ) )
43bibi1d 311 . . . . 5  |-  ( x  =  A  ->  (
( ph  <->  ps )  <->  ( A  e.  { x  |  ph } 
<->  ps ) ) )
54biimpd 199 . . . 4  |-  ( x  =  A  ->  (
( ph  <->  ps )  ->  ( A  e.  { x  |  ph }  <->  ps )
) )
65a2i 13 . . 3  |-  ( ( x  =  A  -> 
( ph  <->  ps ) )  -> 
( x  =  A  ->  ( A  e. 
{ x  |  ph } 
<->  ps ) ) )
76alimi 1568 . 2  |-  ( A. x ( x  =  A  ->  ( ph  <->  ps ) )  ->  A. x
( x  =  A  ->  ( A  e. 
{ x  |  ph } 
<->  ps ) ) )
8 nfcv 2571 . . . 4  |-  F/_ x A
9 nfab1 2573 . . . . . 6  |-  F/_ x { x  |  ph }
109nfel2 2583 . . . . 5  |-  F/ x  A  e.  { x  |  ph }
11 nfv 1629 . . . . 5  |-  F/ x ps
1210, 11nfbi 1856 . . . 4  |-  F/ x
( A  e.  {
x  |  ph }  <->  ps )
13 pm5.5 327 . . . 4  |-  ( x  =  A  ->  (
( x  =  A  ->  ( A  e. 
{ x  |  ph } 
<->  ps ) )  <->  ( A  e.  { x  |  ph } 
<->  ps ) ) )
148, 12, 13spcgf 3023 . . 3  |-  ( A  e.  B  ->  ( A. x ( x  =  A  ->  ( A  e.  { x  |  ph } 
<->  ps ) )  -> 
( A  e.  {
x  |  ph }  <->  ps ) ) )
1514imp 419 . 2  |-  ( ( A  e.  B  /\  A. x ( x  =  A  ->  ( A  e.  { x  |  ph } 
<->  ps ) ) )  ->  ( A  e. 
{ x  |  ph } 
<->  ps ) )
167, 15sylan2 461 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 177    /\ wa 359   A.wal 1549    = wceq 1652    e. wcel 1725   {cab 2421
This theorem is referenced by:  abfmpeld  24058  abfmpel  24059  dfrtrcl2  25140
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-v 2950
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