Theorem opeq12i 4902

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 4899	. 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → ⟨𝐴, 𝐶⟩ = ⟨𝐵, 𝐷⟩)
4	1, 2, 3	mp2an 691	1 ⊢ ⟨𝐴, 𝐶⟩ = ⟨𝐵, 𝐷⟩

Syntax hints:   = wceq 1537  ⟨cop 4654

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655

This theorem is referenced by:  sbcop  5509  elxp6  8064  addcompq  11019  mulcompq  11021  addassnq  11027  mulassnq  11028  distrnq  11030  1lt2nq  11042  axi2m1  11228  om2uzrdg  14007  pzriprng1ALT  21530  pzriprng1  21532  precsexlemcbv  28248  axlowdimlem6  28980  clwlkclwwlkflem  30036  konigsbergvtx  30278  konigsbergiedg  30279  nvop2  30640  nvvop  30641  phop  30850  hhsssh  31301  cshw1s2  32927  rngoi  37859  isdrngo1  37916