Theorem preq12 3471

Description: Equality theorem for unordered pairs. (Contributed by NM, 19-Oct-2012.)

Assertion

Ref	Expression
preq12	|- ( ( A = C /\ B = D ) -> { A , B } = { C , D } )

Proof of Theorem preq12

Step	Hyp	Ref	Expression
1		preq1 3469	. 2 |- ( A = C -> { A , B } = { C , B } )
2		preq2 3470	. 2 |- ( B = D -> { C , B } = { C , D } )
3	1, 2	sylan9eq 2133	1 |- ( ( A = C /\ B = D ) -> { A , B } = { C , D } )

Syntax hints:   -> wi 4   /\ wa 102   = wceq 1284   {cpr 3399

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-un 2977  df-sn 3404  df-pr 3405

This theorem is referenced by:  preq12i  3474  preq12d  3477  preq12b  3562  opthreg  4299  relop  4504