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

Theorem opeq12i 4207
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 4204 . 2  |-  ( ( A  =  B  /\  C  =  D )  -> 
<. A ,  C >.  = 
<. B ,  D >. )
41, 2, 3mp2an 672 1  |-  <. A ,  C >.  =  <. B ,  D >.
Colors of variables: wff setvar class
Syntax hints:    = wceq 1383   <.cop 4020
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-rab 2802  df-v 3097  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021
This theorem is referenced by:  elxp6  6817  addcompq  9331  mulcompq  9333  addassnq  9339  mulassnq  9340  distrnq  9342  1lt2nq  9354  axi2m1  9539  om2uzrdg  12046  axlowdimlem6  24122  rngoi  25254  nvop2  25373  nvvop  25374  phop  25605  hhsssh  26057  isdrngo1  30334
  Copyright terms: Public domain W3C validator