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Theorem preq1 3603
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 3538 . . 3  |-  ( A  =  B  ->  { A }  =  { B } )
21uneq1d 3229 . 2  |-  ( A  =  B  ->  ( { A }  u.  { C } )  =  ( { B }  u.  { C } ) )
3 df-pr 3534 . 2  |-  { A ,  C }  =  ( { A }  u.  { C } )
4 df-pr 3534 . 2  |-  { B ,  C }  =  ( { B }  u.  { C } )
52, 3, 43eqtr4g 2197 1  |-  ( A  =  B  ->  { A ,  C }  =  { B ,  C }
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1331    u. cun 3069   {csn 3527   {cpr 3528
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-un 3075  df-sn 3533  df-pr 3534
This theorem is referenced by:  preq2  3604  preq12  3605  preq1i  3606  preq1d  3609  tpeq1  3612  prnzg  3650  preq12b  3700  preq12bg  3703  opeq1  3708  uniprg  3754  intprg  3807  prexg  4136  opthreg  4474  bdxmet  12696  bj-prexg  13253
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