Theorem opeq1d 3584

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 3578	. 2 ⊢ (𝐴 = 𝐵 → ⟨𝐴, 𝐶⟩ = ⟨𝐵, 𝐶⟩)
3	1, 2	syl 14	1 ⊢ (𝜑 → ⟨𝐴, 𝐶⟩ = ⟨𝐵, 𝐶⟩)

Syntax hints:   → wi 4   = wceq 1285  ⟨cop 3409

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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064

This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-un 2978  df-sn 3412  df-pr 3413  df-op 3415

This theorem is referenced by:  oteq1  3587  oteq2  3588  opth  4000  cbvoprab2  5608  dfplpq2  6606  ltexnqq  6660  nnanq0  6710  addpinq1  6716  prarloclemlo  6746  prarloclem3  6749  prarloclem5  6752  prsrriota  7026  caucvgsrlemfv  7029  caucvgsr  7040  pitonnlem2  7077  pitonn  7078  recidpirq  7088  ax1rid  7105  axrnegex  7107  nntopi  7122  axcaucvglemval  7125  fseq1m1p1  9188  frecuzrdglem  9493  frecuzrdgg  9498  frecuzrdgdomlem  9499  frecuzrdgfunlem  9501  frecuzrdgsuctlem  9505