Theorem opeq2i 4805

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 4802	. 2 ⊢ (𝐴 = 𝐵 → ⟨𝐶, 𝐴⟩ = ⟨𝐶, 𝐵⟩)
3	1, 2	ax-mp 5	1 ⊢ ⟨𝐶, 𝐴⟩ = ⟨𝐶, 𝐵⟩

Syntax hints:   = wceq 1539  ⟨cop 4564

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565

This theorem is referenced by:  fnressn  7012  fressnfv  7014  wfrlem14OLD  8124  seqomlem1  8251  recmulnq  10651  addresr  10825  seqval  13660  ids1  14230  pfx1  14344  pfxccatpfx2  14378  ressinbas  16881  oduval  17922  mgmnsgrpex  18485  sgrpnmndex  18486  efgi0  19241  efgi1  19242  vrgpinv  19290  frgpnabllem1  19389  mat1dimid  21531  uspgr1v1eop  27519  wlk2v2e  28422  avril1  28728  nvop  28939  phop  29081  bnj601  32800  tgrpset  38686  erngset  38741  erngset-rN  38749  zlmodzxzadd  45582  lmod1  45721  lmod1zr  45722  zlmodzxzequa  45725  zlmodzxzequap  45728