Theorem preq1d 4683

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 4677	. 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 4569

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-v 3431  df-un 3894  df-sn 4568  df-pr 4570

This theorem is referenced by:  propeqop  5461  opthwiener  5468  fprg  7109  fprb  7149  fnpr2g  7165  dif1en  9096  dfac2b  10053  symg2bas  19368  crctcshwlkn0lem6  29883  wwlksnredwwlkn  29963  wwlksnextprop  29980  clwwlk1loop  30058  clwlkclwwlklem2fv1  30065  clwlkclwwlklem2fv2  30066  clwlkclwwlklem2a  30068  clwlkclwwlklem3  30071  clwwisshclwwslem  30084  clwwlknlbonbgr1  30109  clwwlkn1  30111  frcond1  30336  frgr1v  30341  nfrgr2v  30342  frgr3v  30345  n4cyclfrgr  30361  2clwwlk2clwwlklem  30416  wopprc  43458  mnurndlem1  44708  grtriclwlk3  48421  isubgr3stgrlem4  48445  gpgedgiov  48541  gpgedg2ov  48542  gpgedg2iv  48543  pgnbgreunbgrlem5lem1  48596  pgnbgreunbgrlem5lem2  48597  pgnbgreunbgrlem5lem3  48598  grlimedgnedg  48607  2arymaptf1  49129  rrx2xpref1o  49194