ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  preqr2g Unicode version

Theorem preqr2g 3782
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 3784. (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 3683 . . 3  |-  { C ,  A }  =  { A ,  C }
2 prcom 3683 . . 3  |-  { C ,  B }  =  { B ,  C }
31, 2eqeq12i 2203 . 2  |-  ( { C ,  A }  =  { C ,  B } 
<->  { A ,  C }  =  { B ,  C } )
4 preqr1g 3781 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( { A ,  C }  =  { B ,  C }  ->  A  =  B ) )
53, 4biimtrid 152 1  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( { C ,  A }  =  { C ,  B }  ->  A  =  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1364    e. wcel 2160   _Vcvv 2752   {cpr 3608
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-un 3148  df-sn 3613  df-pr 3614
This theorem is referenced by:  opth  4255
  Copyright terms: Public domain W3C validator