Theorem preq1d 4720

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

Hypothesis

Ref	Expression
preq1d.1	⊢ (𝜑 → 𝐴 = 𝐵)

Assertion

Ref	Expression
preq1d	⊢ (𝜑 → {𝐴, 𝐶} = {𝐵, 𝐶})

Proof of Theorem preq1d

Step	Hyp	Ref	Expression
1		preq1d.1	. 2 ⊢ (𝜑 → 𝐴 = 𝐵)
2		preq1 4714	. 2 ⊢ (𝐴 = 𝐵 → {𝐴, 𝐶} = {𝐵, 𝐶})
3	1, 2	syl 17	1 ⊢ (𝜑 → {𝐴, 𝐶} = {𝐵, 𝐶})

Syntax hints:   → wi 4   = wceq 1540  {cpr 4608

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-v 3466  df-un 3936  df-sn 4607  df-pr 4609

This theorem is referenced by:  propeqop  5487  opthwiener  5494  fprg  7150  fprb  7191  fnpr2g  7207  dif1en  9179  dif1enOLD  9181  dfac2b  10150  symg2bas  19379  crctcshwlkn0lem6  29802  wwlksnredwwlkn  29882  wwlksnextprop  29899  clwwlk1loop  29974  clwlkclwwlklem2fv1  29981  clwlkclwwlklem2fv2  29982  clwlkclwwlklem2a  29984  clwlkclwwlklem3  29987  clwwisshclwwslem  30000  clwwlknlbonbgr1  30025  clwwlkn1  30027  frcond1  30252  frgr1v  30257  nfrgr2v  30258  frgr3v  30261  n4cyclfrgr  30277  2clwwlk2clwwlklem  30332  wopprc  43021  mnurndlem1  44272  grtriclwlk3  47924  isubgr3stgrlem4  47948  2arymaptf1  48600  rrx2xpref1o  48665