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  4324  cbvoprab2  6086  djuf1olem  7236  dfplpq2  7557  ltexnqq  7611  nnanq0  7661  addpinq1  7667  prarloclemlo  7697  prarloclem3  7700  prarloclem5  7703  prsrriota  7991  caucvgsrlemfv  7994  caucvgsr  8005  pitonnlem2  8050  pitonn  8051  recidpirq  8061  ax1rid  8080  axrnegex  8082  nntopi  8097  axcaucvglemval  8100  fseq1m1p1  10308  frecuzrdglem  10650  frecuzrdgg  10655  frecuzrdgdomlem  10656  frecuzrdgfunlem  10658  frecuzrdgsuctlem  10662  pfxswrd  11259  swrdccat  11288  swrdccat3blem  11292  fsum2dlemstep  11966  fprod2dlemstep  12154  ennnfonelemp1  12998  ennnfonelemnn0  13014  setscomd  13094  imasaddvallemg  13369