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Theorem preq1 3566
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 3504 . . 3  |-  ( A  =  B  ->  { A }  =  { B } )
21uneq1d 3195 . 2  |-  ( A  =  B  ->  ( { A }  u.  { C } )  =  ( { B }  u.  { C } ) )
3 df-pr 3500 . 2  |-  { A ,  C }  =  ( { A }  u.  { C } )
4 df-pr 3500 . 2  |-  { B ,  C }  =  ( { B }  u.  { C } )
52, 3, 43eqtr4g 2172 1  |-  ( A  =  B  ->  { A ,  C }  =  { B ,  C }
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1314    u. cun 3035   {csn 3493   {cpr 3494
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-v 2659  df-un 3041  df-sn 3499  df-pr 3500
This theorem is referenced by:  preq2  3567  preq12  3568  preq1i  3569  preq1d  3572  tpeq1  3575  prnzg  3613  preq12b  3663  preq12bg  3666  opeq1  3671  uniprg  3717  intprg  3770  prexg  4093  opthreg  4431  bdxmet  12490  bj-prexg  12801
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