Theorem opeq2i 4849

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 4846	. 2 ⊢ (𝐴 = 𝐵 → ⟨𝐶, 𝐴⟩ = ⟨𝐶, 𝐵⟩)
3	1, 2	ax-mp 5	1 ⊢ ⟨𝐶, 𝐴⟩ = ⟨𝐶, 𝐵⟩

This proof depends on syntax axioms:   = wceq 1570  ⟨cop 4840

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841

This theorem is used by:  fnressn  7194  fressnfv  7196  seqomlem1  8498  recmulnq  11088  addresr  11262  seqval  14195  ids1  14785  pfx1  14894  pfxccatpfx2  14928  ressinbas  17467  oduval  18512  mgmnsgrpex  19180  sgrpnmndex  19181  efgi0  19985  efgi1  19986  vrgpinv  20034  frgpnabllem1  20138  pzriprng1ALT  21857  mat1dimid  22844  seqsval  28731  uspgr1v1eop  29887  wlk2v2e  30815  avril1  31121  nvop  31335  phop  31477  selvply1rhm0  34269  bnj601  35661  tgrpset  42068  erngset  42123  erngset-rN  42131  nregmodelf1o  46269  stgr0  49331  stgr1  49332  pgnbgreunbgrlem2lem1  49485  pgnbgreunbgrlem2lem2  49486  gpg5edgnedg  49501  zlmodzxzadd  49801  lmod1  49935  lmod1zr  49936  zlmodzxzequa  49939  zlmodzxzequap  49942  cofuoppf  50597  termcfuncval  50998  termcnatval  51004  termolmd  51184