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Theorem elrab3t 2697
Description: Membership in a restricted class abstraction, using implicit substitution. (Closed theorem version of elrab3 2699.) (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 103 . . 3  |-  ( ( A. x ( x  =  A  ->  ( ph 
<->  ps ) )  /\  A  e.  B )  ->  A  e.  B )
2 nfa1 1434 . . . . 5  |-  F/ x A. x ( x  =  A  ->  ( ph  <->  ps ) )
3 nfv 1421 . . . . 5  |-  F/ x  A  e.  B
42, 3nfan 1457 . . . 4  |-  F/ x
( A. x ( x  =  A  -> 
( ph  <->  ps ) )  /\  A  e.  B )
5 simpl 102 . . . . . 6  |-  ( ( A. x ( x  =  A  ->  ( ph 
<->  ps ) )  /\  A  e.  B )  ->  A. x ( x  =  A  ->  ( ph 
<->  ps ) ) )
6519.21bi 1450 . . . . 5  |-  ( ( A. x ( x  =  A  ->  ( ph 
<->  ps ) )  /\  A  e.  B )  ->  ( x  =  A  ->  ( ph  <->  ps )
) )
7 eleq1 2100 . . . . . . . . . 10  |-  ( x  =  A  ->  (
x  e.  B  <->  A  e.  B ) )
87biimparc 283 . . . . . . . . 9  |-  ( ( A  e.  B  /\  x  =  A )  ->  x  e.  B )
98biantrurd 289 . . . . . . . 8  |-  ( ( A  e.  B  /\  x  =  A )  ->  ( ph  <->  ( x  e.  B  /\  ph )
) )
109bibi1d 222 . . . . . . 7  |-  ( ( A  e.  B  /\  x  =  A )  ->  ( ( ph  <->  ps )  <->  ( ( x  e.  B  /\  ph )  <->  ps )
) )
1110pm5.74da 417 . . . . . 6  |-  ( A  e.  B  ->  (
( x  =  A  ->  ( ph  <->  ps )
)  <->  ( x  =  A  ->  ( (
x  e.  B  /\  ph )  <->  ps ) ) ) )
1211adantl 262 . . . . 5  |-  ( ( A. x ( x  =  A  ->  ( ph 
<->  ps ) )  /\  A  e.  B )  ->  ( ( x  =  A  ->  ( ph  <->  ps ) )  <->  ( x  =  A  ->  ( ( x  e.  B  /\  ph )  <->  ps ) ) ) )
136, 12mpbid 135 . . . 4  |-  ( ( A. x ( x  =  A  ->  ( ph 
<->  ps ) )  /\  A  e.  B )  ->  ( x  =  A  ->  ( ( x  e.  B  /\  ph ) 
<->  ps ) ) )
144, 13alrimi 1415 . . 3  |-  ( ( A. x ( x  =  A  ->  ( ph 
<->  ps ) )  /\  A  e.  B )  ->  A. x ( x  =  A  ->  (
( x  e.  B  /\  ph )  <->  ps )
) )
15 elabgt 2684 . . 3  |-  ( ( A  e.  B  /\  A. x ( x  =  A  ->  ( (
x  e.  B  /\  ph )  <->  ps ) ) )  ->  ( A  e. 
{ x  |  ( x  e.  B  /\  ph ) }  <->  ps )
)
161, 14, 15syl2anc 391 . 2  |-  ( ( A. x ( x  =  A  ->  ( ph 
<->  ps ) )  /\  A  e.  B )  ->  ( A  e.  {
x  |  ( x  e.  B  /\  ph ) }  <->  ps ) )
17 df-rab 2315 . . . 4  |-  { x  e.  B  |  ph }  =  { x  |  ( x  e.  B  /\  ph ) }
1817eleq2i 2104 . . 3  |-  ( A  e.  { x  e.  B  |  ph }  <->  A  e.  { x  |  ( x  e.  B  /\  ph ) } )
1918bibi1i 217 . 2  |-  ( ( A  e.  { x  e.  B  |  ph }  <->  ps )  <->  ( A  e. 
{ x  |  ( x  e.  B  /\  ph ) }  <->  ps )
)
2016, 19sylibr 137 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 97    <-> wb 98   A.wal 1241    = wceq 1243    e. wcel 1393   {cab 2026   {crab 2310
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rab 2315  df-v 2559
This theorem is referenced by:  f1oresrab  5329
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