Theorem preq12d 3760

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

Hypotheses

Ref	Expression
preq1d.1	|- ( ph -> A = B )
preq12d.2	|- ( ph -> C = D )

Assertion

Ref	Expression
preq12d	|- ( ph -> { A , C } = { B , D } )

Proof of Theorem preq12d

Step	Hyp	Ref	Expression
1		preq1d.1	. 2 |- ( ph -> A = B )
2		preq12d.2	. 2 |- ( ph -> C = D )
3		preq12 3754	. 2 |- ( ( A = B /\ C = D ) -> { A , C } = { B , D } )
4	1, 2, 3	syl2anc 411	1 |- ( ph -> { A , C } = { B , D } )

Syntax hints:   -> wi 4   = wceq 1398   {cpr 3674

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805  df-un 3205  df-sn 3679  df-pr 3680

This theorem is referenced by:  opeq1  3867  opeq2  3868  xrminrecl  11894  xrminadd  11896  prdsval  13417  xpsfval  13492  xpsval  13496  ring1  14134  xmetxp  15298  xmetxpbl  15299  txmetcnp  15309  hovera  15438  hoverb  15439  hoverlt1  15440  hovergt0  15441  ivthdich  15444  wkslem1  16241  wkslem2  16242  iswlk  16244  2wlklem  16297  isclwwlk  16315  clwwlkccatlem  16321  clwwlkccat  16322  clwwlkn2  16342  clwwlkext2edg  16343  umgr2cwwk2dif  16345  s2elclwwlknon2  16357  clwwlknonex2lem2  16359  clwwlknonex2  16360  eupthseg  16373  eupth2lem3fi  16397