Theorem opeq12i 4836

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

Step	Hyp	Ref	Expression
1		opeq1i.1	. 2 ⊢ 𝐴 = 𝐵
2		opeq12i.2	. 2 ⊢ 𝐶 = 𝐷
3		opeq12 4833	. 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → ⟨𝐴, 𝐶⟩ = ⟨𝐵, 𝐷⟩)
4	1, 2, 3	mp2an 693	1 ⊢ ⟨𝐴, 𝐶⟩ = ⟨𝐵, 𝐷⟩

Syntax hints:   = wceq 1542  ⟨cop 4588

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589

This theorem is referenced by:  sbcop  5445  elxp6  7977  addcompq  10873  mulcompq  10875  addassnq  10881  mulassnq  10882  distrnq  10884  1lt2nq  10896  axi2m1  11082  om2uzrdg  13891  pzriprng1ALT  21466  pzriprng1  21468  precsexlemcbv  28217  axlowdimlem6  29036  clwlkclwwlkflem  30095  konigsbergvtx  30337  konigsbergiedg  30338  nvop2  30700  nvvop  30701  phop  30910  hhsssh  31361  cshw1s2  33057  rngoi  38154  isdrngo1  38211  dfswapf2  49624  swapfcoa  49644  diag1a  49668  funcsetc1o  49860