New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  preqr2 GIF version

Theorem preqr2 4125
 Description: Reverse equality lemma for unordered pairs. If two unordered pairs have the same first element, the second elements are equal. (Contributed by NM, 5-Aug-1993.)
Hypotheses
Ref Expression
preqr2.1 A V
preqr2.2 B V
Assertion
Ref Expression
preqr2 ({C, A} = {C, B} → A = B)

Proof of Theorem preqr2
StepHypRef Expression
1 prcom 3798 . . 3 {C, A} = {A, C}
2 prcom 3798 . . 3 {C, B} = {B, C}
31, 2eqeq12i 2366 . 2 ({C, A} = {C, B} ↔ {A, C} = {B, C})
4 preqr2.1 . . 3 A V
5 preqr2.2 . . 3 B V
64, 5preqr1 4124 . 2 ({A, C} = {B, C} → A = B)
73, 6sylbi 187 1 ({C, A} = {C, B} → A = B)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1642   ∈ wcel 1710  Vcvv 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:  preqr2g  4126  preq12b  4127  opkthg  4131
 Copyright terms: Public domain W3C validator