Theorem preq2 3700

Description: Equality theorem for unordered pairs. (Contributed by NM, 5-Aug-1993.)

Assertion

Ref	Expression
preq2	|- ( A = B -> { C , A } = { C , B } )

Proof of Theorem preq2

Step	Hyp	Ref	Expression
1		preq1 3699	. 2 |- ( A = B -> { A , C } = { B , C } )
2		prcom 3698	. 2 |- { C , A } = { A , C }
3		prcom 3698	. 2 |- { C , B } = { B , C }
4	1, 2, 3	3eqtr4g 2254	1 |- ( A = B -> { C , A } = { C , B } )

Syntax hints:   -> wi 4   = wceq 1364   {cpr 3623

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-un 3161  df-sn 3628  df-pr 3629

This theorem is referenced by:  preq12  3701  preq2i  3703  preq2d  3706  tpeq2  3709  preq12bg  3803  opeq2  3809  uniprg  3854  intprg  3907  prexg  4244  opth  4270  opeqsn  4285  relop  4816  funopg  5292  prfidceq  6989  pr2ne  7259  hashprg  10900  bj-prexg  15557