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

Theorem elabgt 3023
Description: Membership in a class abstraction, using implicit substitution. (Closed theorem version of elabg 3027.) (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 2376 . . . . . . 7  |-  ( x  e.  { x  | 
ph }  <->  ph )
2 eleq1 2448 . . . . . . 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 1565 . 2  |-  ( A. x ( x  =  A  ->  ( ph  <->  ps ) )  ->  A. x
( x  =  A  ->  ( A  e. 
{ x  |  ph } 
<->  ps ) ) )
8 nfcv 2524 . . . 4  |-  F/_ x A
9 nfab1 2526 . . . . . 6  |-  F/_ x { x  |  ph }
109nfel2 2536 . . . . 5  |-  F/ x  A  e.  { x  |  ph }
11 nfv 1626 . . . . 5  |-  F/ x ps
1210, 11nfbi 1846 . . . 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 2975 . . 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 1546    = wceq 1649    e. wcel 1717   {cab 2374
This theorem is referenced by:  abfmpeld  23909  abfmpel  23910  dfrtrcl2  24928
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-v 2902
  Copyright terms: Public domain W3C validator