Theorem opeq12i 4829

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 4826	. 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → ⟨𝐴, 𝐶⟩ = ⟨𝐵, 𝐷⟩)
4	1, 2, 3	mp2an 692	1 ⊢ ⟨𝐴, 𝐶⟩ = ⟨𝐵, 𝐷⟩

Syntax hints:   = wceq 1540  ⟨cop 4583

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-rab 3395  df-v 3438  df-dif 3906  df-un 3908  df-ss 3920  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584

This theorem is referenced by:  sbcop  5432  elxp6  7958  addcompq  10844  mulcompq  10846  addassnq  10852  mulassnq  10853  distrnq  10855  1lt2nq  10867  axi2m1  11053  om2uzrdg  13863  pzriprng1ALT  21403  pzriprng1  21405  precsexlemcbv  28113  axlowdimlem6  28892  clwlkclwwlkflem  29948  konigsbergvtx  30190  konigsbergiedg  30191  nvop2  30552  nvvop  30553  phop  30762  hhsssh  31213  cshw1s2  32903  rngoi  37889  isdrngo1  37946  dfswapf2  49256  swapfcoa  49276  diag1a  49300  funcsetc1o  49492