Theorem preq1d 4675

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 4669	. 2 ⊢ (𝐴 = 𝐵 → {𝐴, 𝐶} = {𝐵, 𝐶})
3	1, 2	syl 17	1 ⊢ (𝜑 → {𝐴, 𝐶} = {𝐵, 𝐶})

Syntax hints:   → wi 4   = wceq 1539  {cpr 4563

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-v 3434  df-un 3892  df-sn 4562  df-pr 4564

This theorem is referenced by:  propeqop  5421  opthwiener  5428  fprg  7027  fprb  7069  fnpr2g  7086  dif1en  8945  dfac2b  9886  symg2bas  19000  crctcshwlkn0lem6  28180  wwlksnredwwlkn  28260  wwlksnextprop  28277  clwwlk1loop  28352  clwlkclwwlklem2fv1  28359  clwlkclwwlklem2fv2  28360  clwlkclwwlklem2a  28362  clwlkclwwlklem3  28365  clwwisshclwwslem  28378  clwwlknlbonbgr1  28403  clwwlkn1  28405  frcond1  28630  frgr1v  28635  nfrgr2v  28636  frgr3v  28639  n4cyclfrgr  28655  2clwwlk2clwwlklem  28710  wopprc  40852  mnurndlem1  41899  isomuspgrlem2d  45283  2arymaptf1  45999  rrx2xpref1o  46064