Theorem opeq1d 3628

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 3622	. 2 ⊢ (𝐴 = 𝐵 → ⟨𝐴, 𝐶⟩ = ⟨𝐵, 𝐶⟩)
3	1, 2	syl 14	1 ⊢ (𝜑 → ⟨𝐴, 𝐶⟩ = ⟨𝐵, 𝐶⟩)

Syntax hints:   → wi 4   = wceq 1289  ⟨cop 3449

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-un 3003  df-sn 3452  df-pr 3453  df-op 3455

This theorem is referenced by:  oteq1  3631  oteq2  3632  opth  4064  cbvoprab2  5721  djuf1olem  6745  dfplpq2  6913  ltexnqq  6967  nnanq0  7017  addpinq1  7023  prarloclemlo  7053  prarloclem3  7056  prarloclem5  7059  prsrriota  7333  caucvgsrlemfv  7336  caucvgsr  7347  pitonnlem2  7384  pitonn  7385  recidpirq  7395  ax1rid  7412  axrnegex  7414  nntopi  7429  axcaucvglemval  7432  fseq1m1p1  9509  frecuzrdglem  9818  frecuzrdgg  9823  frecuzrdgdomlem  9824  frecuzrdgfunlem  9826  frecuzrdgsuctlem  9830  fsum2dlemstep  10828