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Theorem preq1 3684
Description: Equality theorem for unordered pairs. (Contributed by NM, 29-Mar-1998.)
Assertion
Ref Expression
preq1  |-  ( A  =  B  ->  { A ,  C }  =  { B ,  C }
)

Proof of Theorem preq1
StepHypRef Expression
1 sneq 3618 . . 3  |-  ( A  =  B  ->  { A }  =  { B } )
21uneq1d 3303 . 2  |-  ( A  =  B  ->  ( { A }  u.  { C } )  =  ( { B }  u.  { C } ) )
3 df-pr 3614 . 2  |-  { A ,  C }  =  ( { A }  u.  { C } )
4 df-pr 3614 . 2  |-  { B ,  C }  =  ( { B }  u.  { C } )
52, 3, 43eqtr4g 2247 1  |-  ( A  =  B  ->  { A ,  C }  =  { B ,  C }
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1364    u. cun 3142   {csn 3607   {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:  preq2  3685  preq12  3686  preq1i  3687  preq1d  3690  tpeq1  3693  prnzg  3731  preq12b  3785  preq12bg  3788  opeq1  3793  uniprg  3839  intprg  3892  prexg  4226  opthreg  4570  bdxmet  14398  bj-prexg  15060
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