Theorem opeq1d 3764

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 3758	. 2 ⊢ (𝐴 = 𝐵 → ⟨𝐴, 𝐶⟩ = ⟨𝐵, 𝐶⟩)
3	1, 2	syl 14	1 ⊢ (𝜑 → ⟨𝐴, 𝐶⟩ = ⟨𝐵, 𝐶⟩)

Syntax hints:   → wi 4   = wceq 1343  ⟨cop 3579

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-un 3120  df-sn 3582  df-pr 3583  df-op 3585

This theorem is referenced by:  oteq1  3767  oteq2  3768  opth  4215  cbvoprab2  5915  djuf1olem  7018  dfplpq2  7295  ltexnqq  7349  nnanq0  7399  addpinq1  7405  prarloclemlo  7435  prarloclem3  7438  prarloclem5  7441  prsrriota  7729  caucvgsrlemfv  7732  caucvgsr  7743  pitonnlem2  7788  pitonn  7789  recidpirq  7799  ax1rid  7818  axrnegex  7820  nntopi  7835  axcaucvglemval  7838  fseq1m1p1  10030  frecuzrdglem  10346  frecuzrdgg  10351  frecuzrdgdomlem  10352  frecuzrdgfunlem  10354  frecuzrdgsuctlem  10358  fsum2dlemstep  11375  fprod2dlemstep  11563  ennnfonelemp1  12339  ennnfonelemnn0  12355