Theorem preq1d 4744

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 4738	. 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 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:  propeqop  5508  opthwiener  5515  fprg  7156  fprb  7198  fnpr2g  7215  dif1en  9164  dif1enOLD  9166  dfac2b  10129  symg2bas  19303  crctcshwlkn0lem6  29334  wwlksnredwwlkn  29414  wwlksnextprop  29431  clwwlk1loop  29506  clwlkclwwlklem2fv1  29513  clwlkclwwlklem2fv2  29514  clwlkclwwlklem2a  29516  clwlkclwwlklem3  29519  clwwisshclwwslem  29532  clwwlknlbonbgr1  29557  clwwlkn1  29559  frcond1  29784  frgr1v  29789  nfrgr2v  29790  frgr3v  29793  n4cyclfrgr  29809  2clwwlk2clwwlklem  29864  wopprc  42073  mnurndlem1  43344  isomuspgrlem2d  46799  2arymaptf1  47428  rrx2xpref1o  47493