Theorem preq2d 4685

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

Hypothesis

Ref	Expression
preq1d.1	⊢ (𝜑 → 𝐴 = 𝐵)

Assertion

Ref	Expression
preq2d	⊢ (𝜑 → {𝐶, 𝐴} = {𝐶, 𝐵})

Proof of Theorem preq2d

Step	Hyp	Ref	Expression
1		preq1d.1	. 2 ⊢ (𝜑 → 𝐴 = 𝐵)
2		preq2 4679	. 2 ⊢ (𝐴 = 𝐵 → {𝐶, 𝐴} = {𝐶, 𝐵})
3	1, 2	syl 17	1 ⊢ (𝜑 → {𝐶, 𝐴} = {𝐶, 𝐵})

Syntax hints:   → wi 4   = wceq 1542  {cpr 4570

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-v 3432  df-un 3895  df-sn 4569  df-pr 4571

This theorem is referenced by:  opeq2  4818  opthwiener  5462  fprg  7102  fprb  7142  fnprb  7156  fnpr2g  7158  opthreg  9530  fzosplitprm1  13724  s2prop  14860  chnccat  18583  gsumprval  18647  indislem  22975  isconn  23388  hmphindis  23772  wilthlem2  27046  ispth  29804  wwlksnredwwlkn  29978  wwlksnextfun  29981  wwlksnextinj  29982  wwlksnextsurj  29983  wwlksnextbij  29985  clwlkclwwlklem2a1  30077  clwlkclwwlklem2a4  30082  clwlkclwwlklem2  30085  clwwisshclwwslemlem  30098  clwwlkn2  30129  clwwlkf  30132  clwwlknonex2lem1  30192  eupth2lem3lem3  30315  eupth2  30324  frcond1  30351  nfrgr2v  30357  frgr3v  30360  n4cyclfrgr  30376  measxun2  34370  altopthsn  36159  mapdindp4  42183  clnbgrgrimlem  48421  gpgov  48530  gpgprismgriedgdmss  48540  gpgedg2ov  48554  gpgedg2iv  48555  gpg3kgrtriexlem6  48576  gpgprismgr4cycllem3  48585  grlimedgnedg  48619  2arymaptf1  49141  rrx2xpref1o  49206