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Theorem opeq12i 4607
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 4604 . 2 ((𝐴 = 𝐵𝐶 = 𝐷) → ⟨𝐴, 𝐶⟩ = ⟨𝐵, 𝐷⟩)
41, 2, 3mp2an 675 1 𝐴, 𝐶⟩ = ⟨𝐵, 𝐷
Colors of variables: wff setvar class
Syntax hints:   = wceq 1637  cop 4383
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2791
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-clab 2800  df-cleq 2806  df-clel 2809  df-nfc 2944  df-rab 3112  df-v 3400  df-dif 3779  df-un 3781  df-in 3783  df-ss 3790  df-nul 4124  df-if 4287  df-sn 4378  df-pr 4380  df-op 4384
This theorem is referenced by:  elxp6  7435  addcompq  10060  mulcompq  10062  addassnq  10068  mulassnq  10069  distrnq  10071  1lt2nq  10083  axi2m1  10268  om2uzrdg  12982  axlowdimlem6  26047  clwlkclwwlkflem  27153  konigsbergvtx  27425  konigsbergiedg  27426  nvop2  27797  nvvop  27798  phop  28007  hhsssh  28460  rngoi  34011  isdrngo1  34068
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