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Theorem preqr2 3803
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  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 3718 . . 3  |-  { C ,  A }  =  { A ,  C }
2 prcom 3718 . . 3  |-  { C ,  B }  =  { B ,  C }
31, 2eqeq12i 2309 . 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 3802 . 2  |-  ( { A ,  C }  =  { B ,  C }  ->  A  =  B )
73, 6sylbi 187 1  |-  ( { C ,  A }  =  { C ,  B }  ->  A  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696   _Vcvv 2801   {cpr 3654
This theorem is referenced by:  preq12b  3804  opth  4261  opthreg  7335  altopthsn  24567
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-v 2803  df-un 3170  df-sn 3659  df-pr 3660
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