Theorem preq12d 3781

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 3775	. 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 3695

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 2216

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-v 2817  df-un 3218  df-sn 3700  df-pr 3701

This theorem is referenced by:  opeq1  3888  opeq2  3889  xrminrecl  11983  xrminadd  11985  xpsfval  13612  prdsval  14115  xpsval  14143  ring1  14302  xmetxp  15498  xmetxpbl  15499  txmetcnp  15509  hovera  15638  hoverb  15639  hoverlt1  15640  hovergt0  15641  ivthdich  15644  wkslem1  16441  wkslem2  16442  iswlk  16444  2wlklem  16497  isclwwlk  16515  clwwlkccatlem  16521  clwwlkccat  16522  clwwlkn2  16542  clwwlkext2edg  16543  umgr2cwwk2dif  16545  s2elclwwlknon2  16557  clwwlknonex2lem2  16559  clwwlknonex2  16560  eupthseg  16573  eupth2lem3fi  16597