Theorem preq12d 4276

Description: Equality deduction for unordered pairs. (Contributed by NM, 19-Oct-2012.)

Hypotheses

Ref	Expression
preq1d.1	⊢ (𝜑 → 𝐴 = 𝐵)
preq12d.2	⊢ (𝜑 → 𝐶 = 𝐷)

Assertion

Ref	Expression
preq12d	⊢ (𝜑 → {𝐴, 𝐶} = {𝐵, 𝐷})

Proof of Theorem preq12d

Step	Hyp	Ref	Expression
1		preq1d.1	. 2 ⊢ (𝜑 → 𝐴 = 𝐵)
2		preq12d.2	. 2 ⊢ (𝜑 → 𝐶 = 𝐷)
3		preq12 4270	. 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → {𝐴, 𝐶} = {𝐵, 𝐷})
4	1, 2, 3	syl2anc 693	1 ⊢ (𝜑 → {𝐴, 𝐶} = {𝐵, 𝐷})

Syntax hints:   → wi 4   = wceq 1483  {cpr 4179

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-un 3579  df-sn 4178  df-pr 4180

This theorem is referenced by:  opeq1  4402  csbopg  4420  f1oprswap  6180  wunex2  9560  wuncval2  9569  s4prop  13655  wrdlen2  13688  wwlktovf  13699  wwlktovf1  13700  wwlktovfo  13701  wrd2f1tovbij  13703  prdsval  16115  ipoval  17154  frmdval  17388  symg2bas  17818  tusval  22070  tmsval  22286  tmsxpsval  22343  uniiccdif  23346  dchrval  24959  eengv  25859  wkslem1  26503  wkslem2  26504  iswlk  26506  wlkonl1iedg  26561  2wlklem  26563  redwlk  26569  wlkp1lem7  26576  wlkp1lem8  26577  wlkdlem2  26580  upgrwlkdvdelem  26632  usgr2pthlem  26659  usgr2pth  26660  crctcshwlkn0lem4  26705  crctcshwlkn0lem5  26706  crctcshwlkn0lem6  26707  iswwlks  26728  0enwwlksnge1  26749  wlkiswwlks2lem2  26756  wlkiswwlks2lem5  26759  wwlksm1edg  26767  wwlksnred  26787  wwlksnext  26788  wwlksnredwwlkn  26790  wwlksnextproplem2  26805  2wlkdlem10  26831  umgrwwlks2on  26850  rusgrnumwwlkl1  26863  isclwwlks  26880  clwlkclwwlklem2a1  26893  clwlkclwwlklem2fv1  26896  clwlkclwwlklem2a4  26898  clwlkclwwlklem2a  26899  clwlkclwwlklem2  26901  clwlkclwwlk  26903  clwwlkinwwlk  26905  clwwlksn2  26910  clwwlksel  26914  clwwlksf  26915  clwwlksext2edg  26923  wwlksext2clwwlk  26924  wwlksubclwwlks  26925  clwwisshclwwslemlem  26926  clwwisshclwwslem  26927  clwwisshclwws  26928  umgr2cwwk2dif  26941  clwlksfclwwlk  26962  3wlkdlem10  27029  upgr3v3e3cycl  27040  upgr4cycl4dv4e  27045  eupthseg  27066  upgreupthseg  27069  eupth2lem3  27096  nfrgr2v  27136  frgr3vlem1  27137  frgr3vlem2  27138  4cycl2vnunb  27154  extwwlkfablem1  27207  numclwwlkovf2exlem2  27212  numclwwlkovf2ex  27219  disjdifprg  29388  kur14lem1  31188  kur14  31198  tgrpfset  36032  tgrpset  36033  hlhilset  37226  dfac21  37636  mendval  37753  sge0sn  40596  isupwlk  41717  zlmodzxzsub  42138  onsetreclem1  42448