Theorem preq12 3722

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 3720	. 2 |- ( A = C -> { A , B } = { C , B } )
2		preq2 3721	. 2 |- ( B = D -> { C , B } = { C , D } )
3	1, 2	sylan9eq 2260	1 |- ( ( A = C /\ B = D ) -> { A , B } = { C , D } )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   {cpr 3644

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-v 2778  df-un 3178  df-sn 3649  df-pr 3650

This theorem is referenced by:  preq12i  3725  preq12d  3728  ssprsseq  3806  preq12b  3824  elpr2elpr  3830  opthreg  4622  relop  4846  qtopbasss  15108