Theorem preq12d 3792

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 3786	. 2 |- ( ( A = B /\ C = D ) -> { A , C } = { B , D } )
4	1, 2, 3	syl2anc 415	1 |- ( ph -> { A , C } = { B , D } )

Syntax hints:   -> wi 4   = wceq 1402   {cpr 3706

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 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 theorem 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 3711  df-pr 3712

This theorem is referenced by:  opeq1  3899  opeq2  3900  xrminrecl  12017  xrminadd  12019  xpsfval  13646  prdsval  14150  xpsval  14178  ring1  14337  xmetxp  15531  xmetxpbl  15532  txmetcnp  15542  hovera  15671  hoverb  15672  hoverlt1  15673  hovergt0  15674  ivthdich  15677  wkslem1  16475  wkslem2  16476  iswlk  16478  2wlklem  16531  isclwwlk  16549  clwwlkccatlem  16555  clwwlkccat  16556  clwwlkn2  16576  clwwlkext2edg  16577  umgr2cwwk2dif  16579  s2elclwwlknon2  16591  clwwlknonex2lem2  16593  clwwlknonex2  16594  eupthseg  16607  eupth2lem3fi  16631