Theorem preq1d 4693

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 4687	. 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 4581

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 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-v 3440  df-un 3910  df-sn 4580  df-pr 4582

This theorem is referenced by:  propeqop  5454  opthwiener  5461  fprg  7093  fprb  7134  fnpr2g  7150  dif1en  9084  dif1enOLD  9086  dfac2b  10044  symg2bas  19290  crctcshwlkn0lem6  29778  wwlksnredwwlkn  29858  wwlksnextprop  29875  clwwlk1loop  29950  clwlkclwwlklem2fv1  29957  clwlkclwwlklem2fv2  29958  clwlkclwwlklem2a  29960  clwlkclwwlklem3  29963  clwwisshclwwslem  29976  clwwlknlbonbgr1  30001  clwwlkn1  30003  frcond1  30228  frgr1v  30233  nfrgr2v  30234  frgr3v  30237  n4cyclfrgr  30253  2clwwlk2clwwlklem  30308  wopprc  43003  mnurndlem1  44254  grtriclwlk3  47930  isubgr3stgrlem4  47954  gpgedgiov  48050  gpgedg2ov  48051  gpgedg2iv  48052  pgnbgreunbgrlem5lem1  48105  pgnbgreunbgrlem5lem2  48106  pgnbgreunbgrlem5lem3  48107  grlimedgnedg  48116  2arymaptf1  48639  rrx2xpref1o  48704