Theorem opeq2i 4835

Description: Equality inference for ordered pairs. (Contributed by NM, 16-Dec-2006.)

Hypothesis

Ref	Expression
opeq1i.1	⊢ 𝐴 = 𝐵

Assertion

Ref	Expression
opeq2i	⊢ ⟨𝐶, 𝐴⟩ = ⟨𝐶, 𝐵⟩

Proof of Theorem opeq2i

Step	Hyp	Ref	Expression
1		opeq1i.1	. 2 ⊢ 𝐴 = 𝐵
2		opeq2 4832	. 2 ⊢ (𝐴 = 𝐵 → ⟨𝐶, 𝐴⟩ = ⟨𝐶, 𝐵⟩)
3	1, 2	ax-mp 5	1 ⊢ ⟨𝐶, 𝐴⟩ = ⟨𝐶, 𝐵⟩

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:  fnressn  7113  fressnfv  7115  seqomlem1  8391  recmulnq  10887  addresr  11061  seqval  13947  ids1  14533  pfx1  14638  pfxccatpfx2  14672  ressinbas  17184  oduval  18223  mgmnsgrpex  18868  sgrpnmndex  18869  efgi0  19661  efgi1  19662  vrgpinv  19710  frgpnabllem1  19814  dfcnfldOLD  21337  pzriprng1ALT  21463  mat1dimid  22430  seqsval  28296  uspgr1v1eop  29334  wlk2v2e  30244  avril1  30550  nvop  30764  phop  30906  bnj601  35096  tgrpset  41121  erngset  41176  erngset-rN  41184  nregmodelf1o  45371  stgr0  48320  stgr1  48321  pgnbgreunbgrlem2lem1  48474  pgnbgreunbgrlem2lem2  48475  gpg5edgnedg  48490  zlmodzxzadd  48718  lmod1  48852  lmod1zr  48853  zlmodzxzequa  48856  zlmodzxzequap  48859  cofuoppf  49509  termcfuncval  49891  termcnatval  49894  termolmd  50029