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

Theorem preq12i 4666
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 4663 . 2 ((𝐴 = 𝐵𝐶 = 𝐷) → {𝐴, 𝐶} = {𝐵, 𝐷})
41, 2, 3mp2an 688 1 {𝐴, 𝐶} = {𝐵, 𝐷}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1528  {cpr 4559
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-v 3494  df-un 3938  df-sn 4558  df-pr 4560
This theorem is referenced by:  grpbasex  16601  grpplusgx  16602  indistpsx  21546  lgsdir2lem5  25832  wlk2v2elem2  27862  tgrpset  37761  zlmodzxzadd  44334  zlmodzxzequa  44479  zlmodzxzequap  44482
  Copyright terms: Public domain W3C validator