Theorem preq1d 4738

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 4732	. 2 ⊢ (𝐴 = 𝐵 → {𝐴, 𝐶} = {𝐵, 𝐶})
3	1, 2	syl 17	1 ⊢ (𝜑 → {𝐴, 𝐶} = {𝐵, 𝐶})

Syntax hints:   → wi 4   = wceq 1539  {cpr 4627

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1542  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-v 3481  df-un 3955  df-sn 4626  df-pr 4628

This theorem is referenced by:  propeqop  5511  opthwiener  5518  fprg  7174  fprb  7215  fnpr2g  7231  dif1en  9201  dif1enOLD  9203  dfac2b  10172  symg2bas  19411  crctcshwlkn0lem6  29836  wwlksnredwwlkn  29916  wwlksnextprop  29933  clwwlk1loop  30008  clwlkclwwlklem2fv1  30015  clwlkclwwlklem2fv2  30016  clwlkclwwlklem2a  30018  clwlkclwwlklem3  30021  clwwisshclwwslem  30034  clwwlknlbonbgr1  30059  clwwlkn1  30061  frcond1  30286  frgr1v  30291  nfrgr2v  30292  frgr3v  30295  n4cyclfrgr  30311  2clwwlk2clwwlklem  30366  wopprc  43047  mnurndlem1  44305  grtriclwlk3  47917  isubgr3stgrlem4  47941  2arymaptf1  48579  rrx2xpref1o  48644