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Theorem elrab3t 2749
Description: Membership in a restricted class abstraction, using implicit substitution. (Closed theorem version of elrab3 2751.) (Contributed by Thierry Arnoux, 31-Aug-2017.)
Assertion
Ref Expression
elrab3t  |-  ( ( A. x ( x  =  A  ->  ( ph 
<->  ps ) )  /\  A  e.  B )  ->  ( A  e.  {
x  e.  B  |  ph }  <->  ps ) )
Distinct variable groups:    x, A    x, B    ps, x
Allowed substitution hint:    ph( x)

Proof of Theorem elrab3t
StepHypRef Expression
1 simpr 108 . . 3  |-  ( ( A. x ( x  =  A  ->  ( ph 
<->  ps ) )  /\  A  e.  B )  ->  A  e.  B )
2 nfa1 1475 . . . . 5  |-  F/ x A. x ( x  =  A  ->  ( ph  <->  ps ) )
3 nfv 1462 . . . . 5  |-  F/ x  A  e.  B
42, 3nfan 1498 . . . 4  |-  F/ x
( A. x ( x  =  A  -> 
( ph  <->  ps ) )  /\  A  e.  B )
5 simpl 107 . . . . . 6  |-  ( ( A. x ( x  =  A  ->  ( ph 
<->  ps ) )  /\  A  e.  B )  ->  A. x ( x  =  A  ->  ( ph 
<->  ps ) ) )
6519.21bi 1491 . . . . 5  |-  ( ( A. x ( x  =  A  ->  ( ph 
<->  ps ) )  /\  A  e.  B )  ->  ( x  =  A  ->  ( ph  <->  ps )
) )
7 eleq1 2142 . . . . . . . . . 10  |-  ( x  =  A  ->  (
x  e.  B  <->  A  e.  B ) )
87biimparc 293 . . . . . . . . 9  |-  ( ( A  e.  B  /\  x  =  A )  ->  x  e.  B )
98biantrurd 299 . . . . . . . 8  |-  ( ( A  e.  B  /\  x  =  A )  ->  ( ph  <->  ( x  e.  B  /\  ph )
) )
109bibi1d 231 . . . . . . 7  |-  ( ( A  e.  B  /\  x  =  A )  ->  ( ( ph  <->  ps )  <->  ( ( x  e.  B  /\  ph )  <->  ps )
) )
1110pm5.74da 432 . . . . . 6  |-  ( A  e.  B  ->  (
( x  =  A  ->  ( ph  <->  ps )
)  <->  ( x  =  A  ->  ( (
x  e.  B  /\  ph )  <->  ps ) ) ) )
1211adantl 271 . . . . 5  |-  ( ( A. x ( x  =  A  ->  ( ph 
<->  ps ) )  /\  A  e.  B )  ->  ( ( x  =  A  ->  ( ph  <->  ps ) )  <->  ( x  =  A  ->  ( ( x  e.  B  /\  ph )  <->  ps ) ) ) )
136, 12mpbid 145 . . . 4  |-  ( ( A. x ( x  =  A  ->  ( ph 
<->  ps ) )  /\  A  e.  B )  ->  ( x  =  A  ->  ( ( x  e.  B  /\  ph ) 
<->  ps ) ) )
144, 13alrimi 1456 . . 3  |-  ( ( A. x ( x  =  A  ->  ( ph 
<->  ps ) )  /\  A  e.  B )  ->  A. x ( x  =  A  ->  (
( x  e.  B  /\  ph )  <->  ps )
) )
15 elabgt 2736 . . 3  |-  ( ( A  e.  B  /\  A. x ( x  =  A  ->  ( (
x  e.  B  /\  ph )  <->  ps ) ) )  ->  ( A  e. 
{ x  |  ( x  e.  B  /\  ph ) }  <->  ps )
)
161, 14, 15syl2anc 403 . 2  |-  ( ( A. x ( x  =  A  ->  ( ph 
<->  ps ) )  /\  A  e.  B )  ->  ( A  e.  {
x  |  ( x  e.  B  /\  ph ) }  <->  ps ) )
17 df-rab 2358 . . . 4  |-  { x  e.  B  |  ph }  =  { x  |  ( x  e.  B  /\  ph ) }
1817eleq2i 2146 . . 3  |-  ( A  e.  { x  e.  B  |  ph }  <->  A  e.  { x  |  ( x  e.  B  /\  ph ) } )
1918bibi1i 226 . 2  |-  ( ( A  e.  { x  e.  B  |  ph }  <->  ps )  <->  ( A  e. 
{ x  |  ( x  e.  B  /\  ph ) }  <->  ps )
)
2016, 19sylibr 132 1  |-  ( ( A. x ( x  =  A  ->  ( ph 
<->  ps ) )  /\  A  e.  B )  ->  ( A  e.  {
x  e.  B  |  ph }  <->  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103   A.wal 1283    = wceq 1285    e. wcel 1434   {cab 2068   {crab 2353
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-rab 2358  df-v 2604
This theorem is referenced by:  f1oresrab  5355
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