Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  preq1i Structured version   Visualization version   GIF version

Theorem preq1i 4407
 Description: Equality inference for unordered pairs. (Contributed by NM, 19-Oct-2012.)
Hypothesis
Ref Expression
preq1i.1 𝐴 = 𝐵
Assertion
Ref Expression
preq1i {𝐴, 𝐶} = {𝐵, 𝐶}

Proof of Theorem preq1i
StepHypRef Expression
1 preq1i.1 . 2 𝐴 = 𝐵
2 preq1 4404 . 2 (𝐴 = 𝐵 → {𝐴, 𝐶} = {𝐵, 𝐶})
31, 2ax-mp 5 1 {𝐴, 𝐶} = {𝐵, 𝐶}
 Colors of variables: wff setvar class Syntax hints:   = wceq 1631  {cpr 4318 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-v 3353  df-un 3728  df-sn 4317  df-pr 4319 This theorem is referenced by:  funopg  6065  frcond1  27448  n4cyclfrgr  27473  disjdifprg2  29727
 Copyright terms: Public domain W3C validator