MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  opeq12i Structured version   Visualization version   GIF version

Theorem opeq12i 4544
Description: Equality inference for ordered pairs. (Contributed by NM, 16-Dec-2006.) (Proof shortened by Eric Schmidt, 4-Apr-2007.)
Hypotheses
Ref Expression
opeq1i.1 𝐴 = 𝐵
opeq12i.2 𝐶 = 𝐷
Assertion
Ref Expression
opeq12i 𝐴, 𝐶⟩ = ⟨𝐵, 𝐷

Proof of Theorem opeq12i
StepHypRef Expression
1 opeq1i.1 . 2 𝐴 = 𝐵
2 opeq12i.2 . 2 𝐶 = 𝐷
3 opeq12 4541 . 2 ((𝐴 = 𝐵𝐶 = 𝐷) → ⟨𝐴, 𝐶⟩ = ⟨𝐵, 𝐷⟩)
41, 2, 3mp2an 672 1 𝐴, 𝐶⟩ = ⟨𝐵, 𝐷
Colors of variables: wff setvar class
Syntax hints:   = wceq 1631  cop 4322
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-3an 1073  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-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323
This theorem is referenced by:  elxp6  7349  addcompq  9974  mulcompq  9976  addassnq  9982  mulassnq  9983  distrnq  9985  1lt2nq  9997  axi2m1  10182  om2uzrdg  12959  axlowdimlem6  26044  clwlkclwwlkflem  27150  konigsbergvtx  27422  konigsbergiedg  27423  nvop2  27799  nvvop  27800  phop  28009  hhsssh  28462  rngoi  34026  isdrngo1  34083
  Copyright terms: Public domain W3C validator