Theorem preq12d 3796

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 3790	. 2 |- ( ( A = B /\ C = D ) -> { A , C } = { B , D } )
4	1, 2, 3	syl2anc 415	1 |- ( ph -> { A , C } = { B , D } )

This proof depends on syntax axioms:   -> wi 4   = 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:  opeq1  3904  opeq2  3905  xrminrecl  12063  xrminadd  12065  xpsfval  13727  prdsval  14279  xpsval  14307  ring1  14470  psrbagmaxlem1x  15181  psrbagmaxlem1y  15182  xmetxp  15750  xmetxpbl  15751  txmetcnp  15761  hovera  15890  hoverb  15891  hoverlt1  15892  hovergt0  15893  ivthdich  15896  wkslem1  16783  wkslem2  16784  iswlk  16786  2wlklem  16839  isclwwlk  16857  clwwlkccatlem  16863  clwwlkccat  16864  clwwlkn2  16884  clwwlkext2edg  16885  umgr2cwwk2dif  16887  s2elclwwlknon2  16899  clwwlknonex2lem2  16901  clwwlknonex2  16902  eupthseg  16915  eupth2lem3fi  16939