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 1542  {cpr 4631

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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3477  df-un 3954  df-sn 4630  df-pr 4632

This theorem is referenced by:  propeqop  5508  opthwiener  5515  fprg  7153  fprb  7195  fnpr2g  7212  dif1en  9160  dif1enOLD  9162  dfac2b  10125  symg2bas  19260  crctcshwlkn0lem6  29100  wwlksnredwwlkn  29180  wwlksnextprop  29197  clwwlk1loop  29272  clwlkclwwlklem2fv1  29279  clwlkclwwlklem2fv2  29280  clwlkclwwlklem2a  29282  clwlkclwwlklem3  29285  clwwisshclwwslem  29298  clwwlknlbonbgr1  29323  clwwlkn1  29325  frcond1  29550  frgr1v  29555  nfrgr2v  29556  frgr3v  29559  n4cyclfrgr  29575  2clwwlk2clwwlklem  29630  wopprc  41817  mnurndlem1  43088  isomuspgrlem2d  46547  2arymaptf1  47387  rrx2xpref1o  47452