Theorem opeq1d 3862

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 3856	. 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  3865  oteq2  3866  opth  4322  cbvoprab2  6068  djuf1olem  7208  dfplpq2  7529  ltexnqq  7583  nnanq0  7633  addpinq1  7639  prarloclemlo  7669  prarloclem3  7672  prarloclem5  7675  prsrriota  7963  caucvgsrlemfv  7966  caucvgsr  7977  pitonnlem2  8022  pitonn  8023  recidpirq  8033  ax1rid  8052  axrnegex  8054  nntopi  8069  axcaucvglemval  8072  fseq1m1p1  10279  frecuzrdglem  10620  frecuzrdgg  10625  frecuzrdgdomlem  10626  frecuzrdgfunlem  10628  frecuzrdgsuctlem  10632  pfxswrd  11224  swrdccat  11253  swrdccat3blem  11257  fsum2dlemstep  11931  fprod2dlemstep  12119  ennnfonelemp1  12963  ennnfonelemnn0  12979  setscomd  13059  imasaddvallemg  13334