Theorem preq1d 4705

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 4699	. 2 ⊢ (𝐴 = 𝐵 → {𝐴, 𝐶} = {𝐵, 𝐶})
3	1, 2	syl 17	1 ⊢ (𝜑 → {𝐴, 𝐶} = {𝐵, 𝐶})

Syntax hints:   → wi 4   = wceq 1541  {cpr 4593

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-v 3448  df-un 3918  df-sn 4592  df-pr 4594

This theorem is referenced by:  propeqop  5469  opthwiener  5476  fprg  7106  fprb  7148  fnpr2g  7165  dif1en  9111  dif1enOLD  9113  dfac2b  10075  symg2bas  19188  crctcshwlkn0lem6  28823  wwlksnredwwlkn  28903  wwlksnextprop  28920  clwwlk1loop  28995  clwlkclwwlklem2fv1  29002  clwlkclwwlklem2fv2  29003  clwlkclwwlklem2a  29005  clwlkclwwlklem3  29008  clwwisshclwwslem  29021  clwwlknlbonbgr1  29046  clwwlkn1  29048  frcond1  29273  frgr1v  29278  nfrgr2v  29279  frgr3v  29282  n4cyclfrgr  29298  2clwwlk2clwwlklem  29353  wopprc  41412  mnurndlem1  42683  isomuspgrlem2d  46143  2arymaptf1  46859  rrx2xpref1o  46924