MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  opeq12 Structured version   Visualization version   GIF version

Theorem opeq12 4377
Description: Equality theorem for ordered pairs. (Contributed by NM, 28-May-1995.)
Assertion
Ref Expression
opeq12 ((𝐴 = 𝐶𝐵 = 𝐷) → ⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩)

Proof of Theorem opeq12
StepHypRef Expression
1 opeq1 4375 . 2 (𝐴 = 𝐶 → ⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐵⟩)
2 opeq2 4376 . 2 (𝐵 = 𝐷 → ⟨𝐶, 𝐵⟩ = ⟨𝐶, 𝐷⟩)
31, 2sylan9eq 2675 1 ((𝐴 = 𝐶𝐵 = 𝐷) → ⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  cop 4159
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-rab 2916  df-v 3191  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160
This theorem is referenced by:  opeq12i  4380  opeq12d  4383  cbvopab  4688  opth  4910  copsex2t  4922  copsex2g  4923  relop  5237  funopg  5885  fvn0ssdmfun  6311  fsn  6362  fnressn  6385  fmptsng  6394  fmptsnd  6395  tpres  6426  cbvoprab12  6689  eqopi  7154  f1o2ndf1  7237  tposoprab  7340  omeu  7617  brecop  7792  ecovcom  7806  ecovass  7807  ecovdi  7808  xpf1o  8074  addsrmo  9846  mulsrmo  9847  addsrpr  9848  mulsrpr  9849  addcnsr  9908  axcnre  9937  seqeq1  12752  opfi1uzind  13230  opfi1uzindOLD  13236  fsumcnv  14443  fprodcnv  14649  eucalgval2  15229  xpstopnlem1  21535  qustgplem  21847  brabgaf  29286  qqhval2  29832  brsegle  31892  finxpreclem3  32897  dvnprodlem1  39494  funop1  40625  uspgrsprf1  41069
  Copyright terms: Public domain W3C validator