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Theorem preqr2g 3596
Description: Reverse equality lemma for unordered pairs. If two unordered pairs have the same second element, the second elements are equal. Closed form of preqr2 3598. (Contributed by Jim Kingdon, 21-Sep-2018.)
Assertion
Ref Expression
preqr2g  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( { C ,  A }  =  { C ,  B }  ->  A  =  B ) )

Proof of Theorem preqr2g
StepHypRef Expression
1 prcom 3503 . . 3  |-  { C ,  A }  =  { A ,  C }
2 prcom 3503 . . 3  |-  { C ,  B }  =  { B ,  C }
31, 2eqeq12i 2098 . 2  |-  ( { C ,  A }  =  { C ,  B } 
<->  { A ,  C }  =  { B ,  C } )
4 preqr1g 3595 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( { A ,  C }  =  { B ,  C }  ->  A  =  B ) )
53, 4syl5bi 150 1  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( { C ,  A }  =  { C ,  B }  ->  A  =  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1287    e. wcel 1436   _Vcvv 2615   {cpr 3432
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 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-v 2617  df-un 2992  df-sn 3437  df-pr 3438
This theorem is referenced by:  opth  4040
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