Theorem preq1d 4743

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 4737	. 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 4630

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1543  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-v 3475  df-un 3953  df-sn 4629  df-pr 4631

This theorem is referenced by:  propeqop  5507  opthwiener  5514  fprg  7155  fprb  7197  fnpr2g  7214  dif1en  9166  dif1enOLD  9168  dfac2b  10131  symg2bas  19308  crctcshwlkn0lem6  29501  wwlksnredwwlkn  29581  wwlksnextprop  29598  clwwlk1loop  29673  clwlkclwwlklem2fv1  29680  clwlkclwwlklem2fv2  29681  clwlkclwwlklem2a  29683  clwlkclwwlklem3  29686  clwwisshclwwslem  29699  clwwlknlbonbgr1  29724  clwwlkn1  29726  frcond1  29951  frgr1v  29956  nfrgr2v  29957  frgr3v  29960  n4cyclfrgr  29976  2clwwlk2clwwlklem  30031  wopprc  42231  mnurndlem1  43502  isomuspgrlem2d  46957  2arymaptf1  47500  rrx2xpref1o  47565