Theorem preq12 3655

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 3653	. 2 |- ( A = C -> { A , B } = { C , B } )
2		preq2 3654	. 2 |- ( B = D -> { C , B } = { C , D } )
3	1, 2	sylan9eq 2219	1 |- ( ( A = C /\ B = D ) -> { A , B } = { C , D } )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1343   {cpr 3577

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-un 3120  df-sn 3582  df-pr 3583

This theorem is referenced by:  preq12i  3658  preq12d  3661  preq12b  3750  opthreg  4533  relop  4754  qtopbasss  13161