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Theorem preqr1g 3528
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 3530. (Contributed by Jim Kingdon, 21-Sep-2018.)
Assertion
Ref Expression
preqr1g  _V  _V  { ,  C }  { ,  C }

Proof of Theorem preqr1g
StepHypRef Expression
1 prid1g 3465 . . . . . . 7  _V  { ,  C }
2 eleq2 2098 . . . . . . 7  { ,  C }  { ,  C }  { ,  C }  { ,  C }
31, 2syl5ibcom 144 . . . . . 6  _V  { ,  C }  { ,  C }  { ,  C }
4 elprg 3384 . . . . . 6  _V  { ,  C }  C
53, 4sylibd 138 . . . . 5  _V  { ,  C }  { ,  C }  C
65adantr 261 . . . 4  _V  _V  { ,  C }  { ,  C }  C
76imp 115 . . 3  _V  _V  { ,  C }  { ,  C }  C
8 prid1g 3465 . . . . . . 7  _V  { ,  C }
9 eleq2 2098 . . . . . . 7  { ,  C }  { ,  C }  { ,  C }  { ,  C }
108, 9syl5ibrcom 146 . . . . . 6  _V  { ,  C }  { ,  C }  { ,  C }
11 elprg 3384 . . . . . 6  _V  { ,  C }  C
1210, 11sylibd 138 . . . . 5  _V  { ,  C }  { ,  C }  C
1312adantl 262 . . . 4  _V  _V  { ,  C }  { ,  C }  C
1413imp 115 . . 3  _V  _V  { ,  C }  { ,  C }  C
15 eqcom 2039 . . 3
16 eqeq2 2046 . . 3  C  C
177, 14, 15, 16oplem1 881 . 2  _V  _V  { ,  C }  { ,  C }
1817ex 108 1  _V  _V  { ,  C }  { ,  C }
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wo 628   wceq 1242   wcel 1390   _Vcvv 2551   {cpr 3368
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916  df-sn 3373  df-pr 3374
This theorem is referenced by:  preqr2g  3529
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