New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  preqr1 Unicode version

Theorem preqr1 4124
 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 3827 . . . 4
3 eleq2 2414 . . . 4
42, 3mpbii 202 . . 3
51elpr 3751 . . 3
64, 5sylib 188 . 2
7 preqr1.2 . . . . 5
87prid1 3827 . . . 4
9 eleq2 2414 . . . 4
108, 9mpbiri 224 . . 3
117elpr 3751 . . 3
1210, 11sylib 188 . 2
13 eqcom 2355 . 2
14 eqeq2 2362 . 2
156, 12, 13, 14oplem1 930 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wo 357   wceq 1642   wcel 1710  cvv 2859  cpr 3738 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-nin 3211  df-compl 3212  df-un 3214  df-sn 3741  df-pr 3742 This theorem is referenced by:  preqr2  4125
 Copyright terms: Public domain W3C validator