Theorem preq12 3790

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 3788	. 2 |- ( A = C -> { A , B } = { C , B } )
2		preq2 3789	. 2 |- ( B = D -> { C , B } = { C , D } )
3	1, 2	sylan9eq 2291	1 |- ( ( A = C /\ B = D ) -> { A , B } = { C , D } )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   {cpr 3710

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-un 3224  df-sn 3715  df-pr 3716

This theorem is used by:  preq12i  3793  preq12d  3796  ssprsseq  3877  preq12b  3895  elpr2elpr  3901  opthreg  4703  relop  4930  qtopbasss  15622  uspgr2wlkeq  16606  wlkres  16620