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Theorem elabgt 2707
Description: Membership in a class abstraction, using implicit substitution. (Closed theorem version of elabg 2711.) (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 2044 . . . . . . 7  |-  ( x  e.  { x  | 
ph }  <->  ph )
2 eleq1 2116 . . . . . . 7  |-  ( x  =  A  ->  (
x  e.  { x  |  ph }  <->  A  e.  { x  |  ph }
) )
31, 2syl5bbr 187 . . . . . 6  |-  ( x  =  A  ->  ( ph 
<->  A  e.  { x  |  ph } ) )
43bibi1d 226 . . . . 5  |-  ( x  =  A  ->  (
( ph  <->  ps )  <->  ( A  e.  { x  |  ph } 
<->  ps ) ) )
54biimpd 136 . . . 4  |-  ( x  =  A  ->  (
( ph  <->  ps )  ->  ( A  e.  { x  |  ph }  <->  ps )
) )
65a2i 11 . . 3  |-  ( ( x  =  A  -> 
( ph  <->  ps ) )  -> 
( x  =  A  ->  ( A  e. 
{ x  |  ph } 
<->  ps ) ) )
76alimi 1360 . 2  |-  ( A. x ( x  =  A  ->  ( ph  <->  ps ) )  ->  A. x
( x  =  A  ->  ( A  e. 
{ x  |  ph } 
<->  ps ) ) )
8 nfcv 2194 . . . 4  |-  F/_ x A
9 nfab1 2196 . . . . . 6  |-  F/_ x { x  |  ph }
109nfel2 2206 . . . . 5  |-  F/ x  A  e.  { x  |  ph }
11 nfv 1437 . . . . 5  |-  F/ x ps
1210, 11nfbi 1497 . . . 4  |-  F/ x
( A  e.  {
x  |  ph }  <->  ps )
13 pm5.5 235 . . . 4  |-  ( x  =  A  ->  (
( x  =  A  ->  ( A  e. 
{ x  |  ph } 
<->  ps ) )  <->  ( A  e.  { x  |  ph } 
<->  ps ) ) )
148, 12, 13spcgf 2652 . . 3  |-  ( A  e.  B  ->  ( A. x ( x  =  A  ->  ( A  e.  { x  |  ph } 
<->  ps ) )  -> 
( A  e.  {
x  |  ph }  <->  ps ) ) )
1514imp 119 . 2  |-  ( ( A  e.  B  /\  A. x ( x  =  A  ->  ( A  e.  { x  |  ph } 
<->  ps ) ) )  ->  ( A  e. 
{ x  |  ph } 
<->  ps ) )
167, 15sylan2 274 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    /\ wa 101    <-> wb 102   A.wal 1257    = wceq 1259    e. wcel 1409   {cab 2042
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576
This theorem is referenced by:  elrab3t  2720
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