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Theorem preqr1 4204
Description: Reverse equality lemma for unordered pairs. If two unordered pairs have the same second element, the first elements are equal. (Contributed by NM, 18-Oct-1995.)
Hypotheses
Ref Expression
preqr1.1
preqr1.2
Assertion
Ref Expression
preqr1

Proof of Theorem preqr1
StepHypRef Expression
1 preqr1.1 . . . . 5
21prid1 4138 . . . 4
3 eleq2 2530 . . . 4
42, 3mpbii 211 . . 3
51elpr 4047 . . 3
64, 5sylib 196 . 2
7 preqr1.2 . . . . 5
87prid1 4138 . . . 4
9 eleq2 2530 . . . 4
108, 9mpbiri 233 . . 3
117elpr 4047 . . 3
1210, 11sylib 196 . 2
13 eqcom 2466 . 2
14 eqeq2 2472 . 2
156, 12, 13, 14oplem1 964 1
Colors of variables: wff setvar class
Syntax hints:  ->wi 4  \/wo 368  =wceq 1395  e.wcel 1818   cvv 3109  {cpr 4031
This theorem is referenced by:  preqr2  4205  opthwiener  4754  cusgrafilem2  24480  2pthfrgra  25011  wopprc  30972
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-v 3111  df-un 3480  df-sn 4030  df-pr 4032
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