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Theorem elpreqpr 4153
Description: Equality and membership rule for pairs. (Contributed by Scott Fenton, 7-Dec-2020.)
Assertion
Ref Expression
elpreqpr  |-  ( A  e.  { B ,  C }  ->  E. x { B ,  C }  =  { A ,  x } )
Distinct variable groups:    x, A    x, B    x, C

Proof of Theorem elpreqpr
StepHypRef Expression
1 elex 3040 . 2  |-  ( A  e.  { B ,  C }  ->  A  e. 
_V )
2 elpri 3976 . 2  |-  ( A  e.  { B ,  C }  ->  ( A  =  B  \/  A  =  C ) )
3 elpreqprlem 4152 . . . . 5  |-  ( B  e.  _V  ->  E. x { B ,  C }  =  { B ,  x } )
4 eleq1 2537 . . . . . 6  |-  ( A  =  B  ->  ( A  e.  _V  <->  B  e.  _V ) )
5 preq1 4042 . . . . . . . 8  |-  ( A  =  B  ->  { A ,  x }  =  { B ,  x }
)
65eqeq2d 2481 . . . . . . 7  |-  ( A  =  B  ->  ( { B ,  C }  =  { A ,  x } 
<->  { B ,  C }  =  { B ,  x } ) )
76exbidv 1776 . . . . . 6  |-  ( A  =  B  ->  ( E. x { B ,  C }  =  { A ,  x }  <->  E. x { B ,  C }  =  { B ,  x }
) )
84, 7imbi12d 327 . . . . 5  |-  ( A  =  B  ->  (
( A  e.  _V  ->  E. x { B ,  C }  =  { A ,  x }
)  <->  ( B  e. 
_V  ->  E. x { B ,  C }  =  { B ,  x }
) ) )
93, 8mpbiri 241 . . . 4  |-  ( A  =  B  ->  ( A  e.  _V  ->  E. x { B ,  C }  =  { A ,  x }
) )
109impcom 437 . . 3  |-  ( ( A  e.  _V  /\  A  =  B )  ->  E. x { B ,  C }  =  { A ,  x }
)
11 elpreqprlem 4152 . . . . . 6  |-  ( C  e.  _V  ->  E. x { C ,  B }  =  { C ,  x } )
12 prcom 4041 . . . . . . . 8  |-  { C ,  B }  =  { B ,  C }
1312eqeq1i 2476 . . . . . . 7  |-  ( { C ,  B }  =  { C ,  x } 
<->  { B ,  C }  =  { C ,  x } )
1413exbii 1726 . . . . . 6  |-  ( E. x { C ,  B }  =  { C ,  x }  <->  E. x { B ,  C }  =  { C ,  x }
)
1511, 14sylib 201 . . . . 5  |-  ( C  e.  _V  ->  E. x { B ,  C }  =  { C ,  x } )
16 eleq1 2537 . . . . . 6  |-  ( A  =  C  ->  ( A  e.  _V  <->  C  e.  _V ) )
17 preq1 4042 . . . . . . . 8  |-  ( A  =  C  ->  { A ,  x }  =  { C ,  x }
)
1817eqeq2d 2481 . . . . . . 7  |-  ( A  =  C  ->  ( { B ,  C }  =  { A ,  x } 
<->  { B ,  C }  =  { C ,  x } ) )
1918exbidv 1776 . . . . . 6  |-  ( A  =  C  ->  ( E. x { B ,  C }  =  { A ,  x }  <->  E. x { B ,  C }  =  { C ,  x }
) )
2016, 19imbi12d 327 . . . . 5  |-  ( A  =  C  ->  (
( A  e.  _V  ->  E. x { B ,  C }  =  { A ,  x }
)  <->  ( C  e. 
_V  ->  E. x { B ,  C }  =  { C ,  x }
) ) )
2115, 20mpbiri 241 . . . 4  |-  ( A  =  C  ->  ( A  e.  _V  ->  E. x { B ,  C }  =  { A ,  x }
) )
2221impcom 437 . . 3  |-  ( ( A  e.  _V  /\  A  =  C )  ->  E. x { B ,  C }  =  { A ,  x }
)
2310, 22jaodan 802 . 2  |-  ( ( A  e.  _V  /\  ( A  =  B  \/  A  =  C
) )  ->  E. x { B ,  C }  =  { A ,  x } )
241, 2, 23syl2anc 673 1  |-  ( A  e.  { B ,  C }  ->  E. x { B ,  C }  =  { A ,  x } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 375    = wceq 1452   E.wex 1671    e. wcel 1904   _Vcvv 3031   {cpr 3961
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-v 3033  df-dif 3393  df-un 3395  df-nul 3723  df-sn 3960  df-pr 3962
This theorem is referenced by:  elpreqprb  4154
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