Theorem preq1d 4684

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 4678	. 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 4570

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 3432  df-un 3895  df-sn 4569  df-pr 4571

This theorem is referenced by:  propeqop  5455  opthwiener  5462  fprg  7102  fprb  7142  fnpr2g  7158  dif1en  9089  dfac2b  10044  symg2bas  19359  crctcshwlkn0lem6  29898  wwlksnredwwlkn  29978  wwlksnextprop  29995  clwwlk1loop  30073  clwlkclwwlklem2fv1  30080  clwlkclwwlklem2fv2  30081  clwlkclwwlklem2a  30083  clwlkclwwlklem3  30086  clwwisshclwwslem  30099  clwwlknlbonbgr1  30124  clwwlkn1  30126  frcond1  30351  frgr1v  30356  nfrgr2v  30357  frgr3v  30360  n4cyclfrgr  30376  2clwwlk2clwwlklem  30431  wopprc  43476  mnurndlem1  44726  grtriclwlk3  48433  isubgr3stgrlem4  48457  gpgedgiov  48553  gpgedg2ov  48554  gpgedg2iv  48555  pgnbgreunbgrlem5lem1  48608  pgnbgreunbgrlem5lem2  48609  pgnbgreunbgrlem5lem3  48610  grlimedgnedg  48619  2arymaptf1  49141  rrx2xpref1o  49206