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Theorem preq1 3658
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 3592 . . 3  |-  ( A  =  B  ->  { A }  =  { B } )
21uneq1d 3280 . 2  |-  ( A  =  B  ->  ( { A }  u.  { C } )  =  ( { B }  u.  { C } ) )
3 df-pr 3588 . 2  |-  { A ,  C }  =  ( { A }  u.  { C } )
4 df-pr 3588 . 2  |-  { B ,  C }  =  ( { B }  u.  { C } )
52, 3, 43eqtr4g 2228 1  |-  ( A  =  B  ->  { A ,  C }  =  { B ,  C }
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1348    u. cun 3119   {csn 3581   {cpr 3582
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-un 3125  df-sn 3587  df-pr 3588
This theorem is referenced by:  preq2  3659  preq12  3660  preq1i  3661  preq1d  3664  tpeq1  3667  prnzg  3705  preq12b  3755  preq12bg  3758  opeq1  3763  uniprg  3809  intprg  3862  prexg  4194  opthreg  4538  bdxmet  13260  bj-prexg  13911
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