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Theorem preqr2 4194
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 4098 . . 3  |-  { C ,  A }  =  { A ,  C }
2 prcom 4098 . . 3  |-  { C ,  B }  =  { B ,  C }
31, 2eqeq12i 2480 . 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 4193 . 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 1374    e. wcel 1762   _Vcvv 3106   {cpr 4022
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-v 3108  df-un 3474  df-sn 4021  df-pr 4023
This theorem is referenced by:  preq12b  4195  opth  4714  opthreg  8024  usgra2edg  24044  usgraedgreu  24050  nbgraf1olem5  24107  altopthsn  29038  usgvincvadeu  31671  usgvincvadeuALT  31674
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