Theorem opeq1d 3866

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

Hypothesis

Ref	Expression
opeq1d.1	⊢ (𝜑 → 𝐴 = 𝐵)

Assertion

Ref	Expression
opeq1d	⊢ (𝜑 → ⟨𝐴, 𝐶⟩ = ⟨𝐵, 𝐶⟩)

Proof of Theorem opeq1d

Step	Hyp	Ref	Expression
1		opeq1d.1	. 2 ⊢ (𝜑 → 𝐴 = 𝐵)
2		opeq1 3860	. 2 ⊢ (𝐴 = 𝐵 → ⟨𝐴, 𝐶⟩ = ⟨𝐵, 𝐶⟩)
3	1, 2	syl 14	1 ⊢ (𝜑 → ⟨𝐴, 𝐶⟩ = ⟨𝐵, 𝐶⟩)

Syntax hints:   → wi 4   = wceq 1395  ⟨cop 3670

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2802  df-un 3202  df-sn 3673  df-pr 3674  df-op 3676

This theorem is referenced by:  oteq1  3869  oteq2  3870  opth  4327  cbvoprab2  6089  djuf1olem  7246  dfplpq2  7567  ltexnqq  7621  nnanq0  7671  addpinq1  7677  prarloclemlo  7707  prarloclem3  7710  prarloclem5  7713  prsrriota  8001  caucvgsrlemfv  8004  caucvgsr  8015  pitonnlem2  8060  pitonn  8061  recidpirq  8071  ax1rid  8090  axrnegex  8092  nntopi  8107  axcaucvglemval  8110  fseq1m1p1  10323  frecuzrdglem  10666  frecuzrdgg  10671  frecuzrdgdomlem  10672  frecuzrdgfunlem  10674  frecuzrdgsuctlem  10678  pfxswrd  11280  swrdccat  11309  swrdccat3blem  11313  fsum2dlemstep  11988  fprod2dlemstep  12176  ennnfonelemp1  13020  ennnfonelemnn0  13036  setscomd  13116  imasaddvallemg  13391