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Theorem preqr2 4191
Description: Reverse equality lemma for unordered pairs. If two unordered pairs have the same first element, the second elements are equal. (Contributed by NM, 15-Jul-1993.)
Hypotheses
Ref Expression
preqr2.1  |-  A  e. 
_V
preqr2.2  |-  B  e. 
_V
Assertion
Ref Expression
preqr2  |-  ( { C ,  A }  =  { C ,  B }  ->  A  =  B )

Proof of Theorem preqr2
StepHypRef Expression
1 prcom 4094 . . 3  |-  { C ,  A }  =  { A ,  C }
2 prcom 4094 . . 3  |-  { C ,  B }  =  { B ,  C }
31, 2eqeq12i 2474 . 2  |-  ( { C ,  A }  =  { C ,  B } 
<->  { A ,  C }  =  { B ,  C } )
4 preqr2.1 . . 3  |-  A  e. 
_V
5 preqr2.2 . . 3  |-  B  e. 
_V
64, 5preqr1 4190 . 2  |-  ( { A ,  C }  =  { B ,  C }  ->  A  =  B )
73, 6sylbi 195 1  |-  ( { C ,  A }  =  { C ,  B }  ->  A  =  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1398    e. wcel 1823   _Vcvv 3106   {cpr 4018
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-v 3108  df-un 3466  df-sn 4017  df-pr 4019
This theorem is referenced by:  preq12b  4192  opth  4711  opthreg  8026  usgra2edg  24584  usgraedgreu  24590  nbgraf1olem5  24647  altopthsn  29839  usgvincvadeu  32777  usgvincvadeuALT  32780
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