Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  preqr1 Unicode version

Theorem preqr1 3568
 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 3506 . . . 4
3 eleq2 2143 . . . 4
42, 3mpbii 146 . . 3
51elpr 3427 . . 3
64, 5sylib 120 . 2
7 preqr1.2 . . . . 5
87prid1 3506 . . . 4
9 eleq2 2143 . . . 4
108, 9mpbiri 166 . . 3
117elpr 3427 . . 3
1210, 11sylib 120 . 2
13 eqcom 2084 . 2
14 eqeq2 2091 . 2
156, 12, 13, 14oplem1 917 1
 Colors of variables: wff set class Syntax hints:   wi 4   wo 662   wceq 1285   wcel 1434  cvv 2602  cpr 3407 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064 This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-un 2978  df-sn 3412  df-pr 3413 This theorem is referenced by:  preqr2  3569
 Copyright terms: Public domain W3C validator