Theorem opeq2i 4788

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 4785	. 2 ⊢ (𝐴 = 𝐵 → ⟨𝐶, 𝐴⟩ = ⟨𝐶, 𝐵⟩)
3	1, 2	ax-mp 5	1 ⊢ ⟨𝐶, 𝐴⟩ = ⟨𝐶, 𝐵⟩

Syntax hints:   = wceq 1543  ⟨cop 4547

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-ext 2708

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548

This theorem is referenced by:  fnressn  6973  fressnfv  6975  wfrlem14  8068  seqomlem1  8186  recmulnq  10578  addresr  10752  seqval  13585  ids1  14154  pfx1  14268  pfxccatpfx2  14302  ressinbas  16797  oduval  17796  mgmnsgrpex  18358  sgrpnmndex  18359  efgi0  19110  efgi1  19111  vrgpinv  19159  frgpnabllem1  19258  mat1dimid  21371  uspgr1v1eop  27337  wlk2v2e  28240  avril1  28546  nvop  28757  phop  28899  bnj601  32613  tgrpset  38496  erngset  38551  erngset-rN  38559  zlmodzxzadd  45367  lmod1  45506  lmod1zr  45507  zlmodzxzequa  45510  zlmodzxzequap  45513