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Theorem elrab3t 3181
Description: Membership in a restricted class abstraction, using implicit substitution. (Closed theorem version of elrab3 3183.) (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 df-rab 2741 . . 3  |-  { x  e.  B  |  ph }  =  { x  |  ( x  e.  B  /\  ph ) }
21eleq2i 2460 . 2  |-  ( A  e.  { x  e.  B  |  ph }  <->  A  e.  { x  |  ( x  e.  B  /\  ph ) } )
3 simpr 459 . . 3  |-  ( ( A. x ( x  =  A  ->  ( ph 
<->  ps ) )  /\  A  e.  B )  ->  A  e.  B )
4 nfa1 1905 . . . . 5  |-  F/ x A. x ( x  =  A  ->  ( ph  <->  ps ) )
5 nfv 1715 . . . . 5  |-  F/ x  A  e.  B
64, 5nfan 1936 . . . 4  |-  F/ x
( A. x ( x  =  A  -> 
( ph  <->  ps ) )  /\  A  e.  B )
7 simpl 455 . . . . . 6  |-  ( ( A. x ( x  =  A  ->  ( ph 
<->  ps ) )  /\  A  e.  B )  ->  A. x ( x  =  A  ->  ( ph 
<->  ps ) ) )
8719.21bi 1877 . . . . 5  |-  ( ( A. x ( x  =  A  ->  ( ph 
<->  ps ) )  /\  A  e.  B )  ->  ( x  =  A  ->  ( ph  <->  ps )
) )
9 eleq1 2454 . . . . . . . . . 10  |-  ( x  =  A  ->  (
x  e.  B  <->  A  e.  B ) )
109biimparc 485 . . . . . . . . 9  |-  ( ( A  e.  B  /\  x  =  A )  ->  x  e.  B )
1110biantrurd 506 . . . . . . . 8  |-  ( ( A  e.  B  /\  x  =  A )  ->  ( ph  <->  ( x  e.  B  /\  ph )
) )
1211bibi1d 317 . . . . . . 7  |-  ( ( A  e.  B  /\  x  =  A )  ->  ( ( ph  <->  ps )  <->  ( ( x  e.  B  /\  ph )  <->  ps )
) )
1312pm5.74da 685 . . . . . 6  |-  ( A  e.  B  ->  (
( x  =  A  ->  ( ph  <->  ps )
)  <->  ( x  =  A  ->  ( (
x  e.  B  /\  ph )  <->  ps ) ) ) )
1413adantl 464 . . . . 5  |-  ( ( A. x ( x  =  A  ->  ( ph 
<->  ps ) )  /\  A  e.  B )  ->  ( ( x  =  A  ->  ( ph  <->  ps ) )  <->  ( x  =  A  ->  ( ( x  e.  B  /\  ph )  <->  ps ) ) ) )
158, 14mpbid 210 . . . 4  |-  ( ( A. x ( x  =  A  ->  ( ph 
<->  ps ) )  /\  A  e.  B )  ->  ( x  =  A  ->  ( ( x  e.  B  /\  ph ) 
<->  ps ) ) )
166, 15alrimi 1885 . . 3  |-  ( ( A. x ( x  =  A  ->  ( ph 
<->  ps ) )  /\  A  e.  B )  ->  A. x ( x  =  A  ->  (
( x  e.  B  /\  ph )  <->  ps )
) )
17 elabgt 3168 . . 3  |-  ( ( A  e.  B  /\  A. x ( x  =  A  ->  ( (
x  e.  B  /\  ph )  <->  ps ) ) )  ->  ( A  e. 
{ x  |  ( x  e.  B  /\  ph ) }  <->  ps )
)
183, 16, 17syl2anc 659 . 2  |-  ( ( A. x ( x  =  A  ->  ( ph 
<->  ps ) )  /\  A  e.  B )  ->  ( A  e.  {
x  |  ( x  e.  B  /\  ph ) }  <->  ps ) )
192, 18syl5bb 257 1  |-  ( ( A. x ( x  =  A  ->  ( ph 
<->  ps ) )  /\  A  e.  B )  ->  ( A  e.  {
x  e.  B  |  ph }  <->  ps ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367   A.wal 1397    = wceq 1399    e. wcel 1826   {cab 2367   {crab 2736
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360
This theorem depends on definitions:  df-bi 185  df-an 369  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-rab 2741  df-v 3036
This theorem is referenced by:  f1oresrab  5965
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