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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
Distinct variable groups:   ,   ,   ,

Proof of Theorem elpreqpr
StepHypRef Expression
1 elex 3040 . 2
2 elpri 3976 . 2
3 elpreqprlem 4152 . . . . 5
4 eleq1 2537 . . . . . 6
5 preq1 4042 . . . . . . . 8
65eqeq2d 2481 . . . . . . 7
76exbidv 1776 . . . . . 6
84, 7imbi12d 327 . . . . 5
93, 8mpbiri 241 . . . 4
109impcom 437 . . 3
11 elpreqprlem 4152 . . . . . 6
12 prcom 4041 . . . . . . . 8
1312eqeq1i 2476 . . . . . . 7
1413exbii 1726 . . . . . 6
1511, 14sylib 201 . . . . 5
16 eleq1 2537 . . . . . 6
17 preq1 4042 . . . . . . . 8
1817eqeq2d 2481 . . . . . . 7
1918exbidv 1776 . . . . . 6
2016, 19imbi12d 327 . . . . 5
2115, 20mpbiri 241 . . . 4
2221impcom 437 . . 3
2310, 22jaodan 802 . 2
241, 2, 23syl2anc 673 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wo 375   wceq 1452  wex 1671   wcel 1904  cvv 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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