Theorem preq12d 3754

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 3748	. 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 1395   {cpr 3668

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2802  df-un 3202  df-sn 3673  df-pr 3674

This theorem is referenced by:  opeq1  3860  opeq2  3861  xrminrecl  11824  xrminadd  11826  prdsval  13346  xpsfval  13421  xpsval  13425  ring1  14062  xmetxp  15221  xmetxpbl  15222  txmetcnp  15232  hovera  15361  hoverb  15362  hoverlt1  15363  hovergt0  15364  ivthdich  15367  wkslem1  16117  wkslem2  16118  iswlk  16120  2wlklem  16171  isclwwlk  16189  clwwlkccatlem  16195  clwwlkccat  16196  clwwlkn2  16216  clwwlkext2edg  16217  umgr2cwwk2dif  16219  s2elclwwlknon2  16231  clwwlknonex2lem2  16233  clwwlknonex2  16234  eupthseg  16247