Theorem opeq1i 4857

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

Step	Hyp	Ref	Expression
1		opeq1i.1	. 2 ⊢ 𝐴 = 𝐵
2		opeq1 4854	. 2 ⊢ (𝐴 = 𝐵 → ⟨𝐴, 𝐶⟩ = ⟨𝐵, 𝐶⟩)
3	1, 2	ax-mp 5	1 ⊢ ⟨𝐴, 𝐶⟩ = ⟨𝐵, 𝐶⟩

Syntax hints:   = wceq 1540  ⟨cop 4612

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 2708

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 2715  df-cleq 2728  df-clel 2810  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613

This theorem is referenced by:  axi2m1  11178  s3tpop  14933  2strop  17255  grpbasex  17311  grpplusgx  17312  mat1dimelbas  22414  mat1dim0  22416  mat1dimid  22417  mat1dimscm  22418  mat1dimmul  22419  indistpsx  22953  nosupcbv  27671  noinfcbv  27686  setsiedg  29020  cusgrsize  29439  mapfzcons  42706