Theorem opeq2i 4821

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 4818	. 2 ⊢ (𝐴 = 𝐵 → ⟨𝐶, 𝐴⟩ = ⟨𝐶, 𝐵⟩)
3	1, 2	ax-mp 5	1 ⊢ ⟨𝐶, 𝐴⟩ = ⟨𝐶, 𝐵⟩

Syntax hints:   = wceq 1542  ⟨cop 4574

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 3391  df-v 3432  df-dif 3893  df-un 3895  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575

This theorem is referenced by:  fnressn  7106  fressnfv  7108  seqomlem1  8383  recmulnq  10881  addresr  11055  seqval  13968  ids1  14554  pfx1  14659  pfxccatpfx2  14693  ressinbas  17209  oduval  18248  mgmnsgrpex  18896  sgrpnmndex  18897  efgi0  19689  efgi1  19690  vrgpinv  19738  frgpnabllem1  19842  dfcnfldOLD  21363  pzriprng1ALT  21489  mat1dimid  22452  seqsval  28297  uspgr1v1eop  29335  wlk2v2e  30245  avril1  30551  nvop  30765  phop  30907  bnj601  35081  tgrpset  41208  erngset  41263  erngset-rN  41271  nregmodelf1o  45463  stgr0  48451  stgr1  48452  pgnbgreunbgrlem2lem1  48605  pgnbgreunbgrlem2lem2  48606  gpg5edgnedg  48621  zlmodzxzadd  48849  lmod1  48983  lmod1zr  48984  zlmodzxzequa  48987  zlmodzxzequap  48990  cofuoppf  49640  termcfuncval  50022  termcnatval  50025  termolmd  50160