Theorem preq2d 4745

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 4739	. 2 ⊢ (𝐴 = 𝐵 → {𝐶, 𝐴} = {𝐶, 𝐵})
3	1, 2	syl 17	1 ⊢ (𝜑 → {𝐶, 𝐴} = {𝐶, 𝐵})

Syntax hints:   → wi 4   = wceq 1539  {cpr 4631

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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-tru 1542  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-v 3474  df-un 3954  df-sn 4630  df-pr 4632

This theorem is referenced by:  opeq2  4875  opthwiener  5515  fprg  7156  fprb  7198  fnprb  7213  fnpr2g  7215  opthreg  9617  fzosplitprm1  13748  s2prop  14864  gsumprval  18615  indislem  22725  isconn  23139  hmphindis  23523  wilthlem2  26807  ispth  29245  wwlksnredwwlkn  29414  wwlksnextfun  29417  wwlksnextinj  29418  wwlksnextsurj  29419  wwlksnextbij  29421  clwlkclwwlklem2a1  29510  clwlkclwwlklem2a4  29515  clwlkclwwlklem2  29518  clwwisshclwwslemlem  29531  clwwlkn2  29562  clwwlkf  29565  clwwlknonex2lem1  29625  eupth2lem3lem3  29748  eupth2  29757  frcond1  29784  nfrgr2v  29790  frgr3v  29793  n4cyclfrgr  29809  measxun2  33504  altopthsn  35235  mapdindp4  40899  isomuspgrlem2d  46799  2arymaptf1  47428  rrx2xpref1o  47493