MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  preq12i Structured version   Visualization version   GIF version

Theorem preq12i 4492
Description: Equality inference for unordered pairs. (Contributed by NM, 19-Oct-2012.)
Hypotheses
Ref Expression
preq1i.1 𝐴 = 𝐵
preq12i.2 𝐶 = 𝐷
Assertion
Ref Expression
preq12i {𝐴, 𝐶} = {𝐵, 𝐷}

Proof of Theorem preq12i
StepHypRef Expression
1 preq1i.1 . 2 𝐴 = 𝐵
2 preq12i.2 . 2 𝐶 = 𝐷
3 preq12 4489 . 2 ((𝐴 = 𝐵𝐶 = 𝐷) → {𝐴, 𝐶} = {𝐵, 𝐷})
41, 2, 3mp2an 685 1 {𝐴, 𝐶} = {𝐵, 𝐷}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1658  {cpr 4400
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-ext 2804
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-v 3417  df-un 3804  df-sn 4399  df-pr 4401
This theorem is referenced by:  grpbasex  16354  grpplusgx  16355  indistpsx  21186  lgsdir2lem5  25468  wlk2v2elem2  27533  tgrpset  36821  zlmodzxzadd  42984  zlmodzxzequa  43133  zlmodzxzequap  43136
  Copyright terms: Public domain W3C validator