Theorem preq2d 4684

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 4678	. 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 4569

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 2708

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 2715  df-cleq 2728  df-clel 2811  df-v 3431  df-un 3894  df-sn 4568  df-pr 4570

This theorem is referenced by:  opeq2  4817  opthwiener  5468  fprg  7109  fprb  7149  fnprb  7163  fnpr2g  7165  opthreg  9539  fzosplitprm1  13733  s2prop  14869  chnccat  18592  gsumprval  18656  indislem  22965  isconn  23378  hmphindis  23762  wilthlem2  27032  ispth  29789  wwlksnredwwlkn  29963  wwlksnextfun  29966  wwlksnextinj  29967  wwlksnextsurj  29968  wwlksnextbij  29970  clwlkclwwlklem2a1  30062  clwlkclwwlklem2a4  30067  clwlkclwwlklem2  30070  clwwisshclwwslemlem  30083  clwwlkn2  30114  clwwlkf  30117  clwwlknonex2lem1  30177  eupth2lem3lem3  30300  eupth2  30309  frcond1  30336  nfrgr2v  30342  frgr3v  30345  n4cyclfrgr  30361  measxun2  34354  altopthsn  36143  mapdindp4  42169  clnbgrgrimlem  48409  gpgov  48518  gpgprismgriedgdmss  48528  gpgedg2ov  48542  gpgedg2iv  48543  gpg3kgrtriexlem6  48564  gpgprismgr4cycllem3  48573  grlimedgnedg  48607  2arymaptf1  49129  rrx2xpref1o  49194