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  11851  xrminadd  11853  prdsval  13374  xpsfval  13449  xpsval  13453  ring1  14091  xmetxp  15250  xmetxpbl  15251  txmetcnp  15261  hovera  15390  hoverb  15391  hoverlt1  15392  hovergt0  15393  ivthdich  15396  wkslem1  16190  wkslem2  16191  iswlk  16193  2wlklem  16246  isclwwlk  16264  clwwlkccatlem  16270  clwwlkccat  16271  clwwlkn2  16291  clwwlkext2edg  16292  umgr2cwwk2dif  16294  s2elclwwlknon2  16306  clwwlknonex2lem2  16308  clwwlknonex2  16309  eupthseg  16322  eupth2lem3fi  16346