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

Theorem opeq12i 4339
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 4336 . 2 ((𝐴 = 𝐵𝐶 = 𝐷) → ⟨𝐴, 𝐶⟩ = ⟨𝐵, 𝐷⟩)
41, 2, 3mp2an 703 1 𝐴, 𝐶⟩ = ⟨𝐵, 𝐷
Colors of variables: wff setvar class
Syntax hints:   = wceq 1474  cop 4130
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-rab 2904  df-v 3174  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131
This theorem is referenced by:  elxp6  7069  addcompq  9629  mulcompq  9631  addassnq  9637  mulassnq  9638  distrnq  9640  1lt2nq  9652  axi2m1  9837  om2uzrdg  12575  axlowdimlem6  25573  nvop2  26659  nvvop  26660  phop  26891  hhsssh  27344  rngoi  32692  isdrngo1  32749  konigsbergvtx  41436  konigsbergiedg  41437
  Copyright terms: Public domain W3C validator