MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  opeq1i Structured version   Visualization version   GIF version

Theorem opeq1i 4380
Description: Equality inference for ordered pairs. (Contributed by NM, 16-Dec-2006.)
Hypothesis
Ref Expression
opeq1i.1 𝐴 = 𝐵
Assertion
Ref Expression
opeq1i 𝐴, 𝐶⟩ = ⟨𝐵, 𝐶

Proof of Theorem opeq1i
StepHypRef Expression
1 opeq1i.1 . 2 𝐴 = 𝐵
2 opeq1 4377 . 2 (𝐴 = 𝐵 → ⟨𝐴, 𝐶⟩ = ⟨𝐵, 𝐶⟩)
31, 2ax-mp 5 1 𝐴, 𝐶⟩ = ⟨𝐵, 𝐶
Colors of variables: wff setvar class
Syntax hints:   = wceq 1480  cop 4161
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 2917  df-v 3192  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162
This theorem is referenced by:  axi2m1  9940  s3tpop  13606  strlemor1OLD  15909  2strstr1  15926  2strop1  15928  grpbasex  15934  grpplusgx  15935  mat1dimelbas  20217  mat1dim0  20219  mat1dimid  20220  mat1dimscm  20221  mat1dimmul  20222  indistpsx  20754  setsiedg  25862  cusgrsize  26271  mapfzcons  36798
  Copyright terms: Public domain W3C validator