Theorem preq1d 4697

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 4691	. 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 4583

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 2709

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 2716  df-cleq 2729  df-clel 2812  df-v 3443  df-un 3907  df-sn 4582  df-pr 4584

This theorem is referenced by:  propeqop  5456  opthwiener  5463  fprg  7102  fprb  7142  fnpr2g  7158  dif1en  9090  dfac2b  10045  symg2bas  19326  crctcshwlkn0lem6  29871  wwlksnredwwlkn  29951  wwlksnextprop  29968  clwwlk1loop  30046  clwlkclwwlklem2fv1  30053  clwlkclwwlklem2fv2  30054  clwlkclwwlklem2a  30056  clwlkclwwlklem3  30059  clwwisshclwwslem  30072  clwwlknlbonbgr1  30097  clwwlkn1  30099  frcond1  30324  frgr1v  30329  nfrgr2v  30330  frgr3v  30333  n4cyclfrgr  30349  2clwwlk2clwwlklem  30404  wopprc  43308  mnurndlem1  44558  grtriclwlk3  48227  isubgr3stgrlem4  48251  gpgedgiov  48347  gpgedg2ov  48348  gpgedg2iv  48349  pgnbgreunbgrlem5lem1  48402  pgnbgreunbgrlem5lem2  48403  pgnbgreunbgrlem5lem3  48404  grlimedgnedg  48413  2arymaptf1  48935  rrx2xpref1o  49000