Theorem preq12d 3756

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 3750	. 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 1397   {cpr 3670

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-un 3204  df-sn 3675  df-pr 3676

This theorem is referenced by:  opeq1  3862  opeq2  3863  xrminrecl  11833  xrminadd  11835  prdsval  13355  xpsfval  13430  xpsval  13434  ring1  14071  xmetxp  15230  xmetxpbl  15231  txmetcnp  15241  hovera  15370  hoverb  15371  hoverlt1  15372  hovergt0  15373  ivthdich  15376  wkslem1  16170  wkslem2  16171  iswlk  16173  2wlklem  16226  isclwwlk  16244  clwwlkccatlem  16250  clwwlkccat  16251  clwwlkn2  16271  clwwlkext2edg  16272  umgr2cwwk2dif  16274  s2elclwwlknon2  16286  clwwlknonex2lem2  16288  clwwlknonex2  16289  eupthseg  16302