Theorem opeq1d 3863

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 3857	. 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 3669

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 2801  df-un 3201  df-sn 3672  df-pr 3673  df-op 3675

This theorem is referenced by:  oteq1  3866  oteq2  3867  opth  4323  cbvoprab2  6083  djuf1olem  7228  dfplpq2  7549  ltexnqq  7603  nnanq0  7653  addpinq1  7659  prarloclemlo  7689  prarloclem3  7692  prarloclem5  7695  prsrriota  7983  caucvgsrlemfv  7986  caucvgsr  7997  pitonnlem2  8042  pitonn  8043  recidpirq  8053  ax1rid  8072  axrnegex  8074  nntopi  8089  axcaucvglemval  8092  fseq1m1p1  10299  frecuzrdglem  10641  frecuzrdgg  10646  frecuzrdgdomlem  10647  frecuzrdgfunlem  10649  frecuzrdgsuctlem  10653  pfxswrd  11246  swrdccat  11275  swrdccat3blem  11279  fsum2dlemstep  11953  fprod2dlemstep  12141  ennnfonelemp1  12985  ennnfonelemnn0  13001  setscomd  13081  imasaddvallemg  13356