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Theorem preqr1g 3537
 Description: Reverse equality lemma for unordered pairs. If two unordered pairs have the same second element, the first elements are equal. Closed form of preqr1 3539. (Contributed by Jim Kingdon, 21-Sep-2018.)
Assertion
Ref Expression
preqr1g

Proof of Theorem preqr1g
StepHypRef Expression
1 prid1g 3474 . . . . . . 7
2 eleq2 2101 . . . . . . 7
31, 2syl5ibcom 144 . . . . . 6
4 elprg 3395 . . . . . 6
53, 4sylibd 138 . . . . 5
65adantr 261 . . . 4
76imp 115 . . 3
8 prid1g 3474 . . . . . . 7
9 eleq2 2101 . . . . . . 7
108, 9syl5ibrcom 146 . . . . . 6
11 elprg 3395 . . . . . 6
1210, 11sylibd 138 . . . . 5
1312adantl 262 . . . 4
1413imp 115 . . 3
15 eqcom 2042 . . 3
16 eqeq2 2049 . . 3
177, 14, 15, 16oplem1 882 . 2
1817ex 108 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 97   wo 629   wceq 1243   wcel 1393  cvv 2557  cpr 3376 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-v 2559  df-un 2922  df-sn 3381  df-pr 3382 This theorem is referenced by:  preqr2g  3538
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