Theorem preq12d 3776

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 3770	. 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 1398   {cpr 3690

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-v 2815  df-un 3215  df-sn 3695  df-pr 3696

This theorem is referenced by:  opeq1  3883  opeq2  3884  xrminrecl  11958  xrminadd  11960  prdsval  13486  xpsfval  13561  xpsval  13565  ring1  14203  xmetxp  15372  xmetxpbl  15373  txmetcnp  15383  hovera  15512  hoverb  15513  hoverlt1  15514  hovergt0  15515  ivthdich  15518  wkslem1  16315  wkslem2  16316  iswlk  16318  2wlklem  16371  isclwwlk  16389  clwwlkccatlem  16395  clwwlkccat  16396  clwwlkn2  16416  clwwlkext2edg  16417  umgr2cwwk2dif  16419  s2elclwwlknon2  16431  clwwlknonex2lem2  16433  clwwlknonex2  16434  eupthseg  16447  eupth2lem3fi  16471