Theorem opeq1d 3908

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 3902	. 2 ⊢ (𝐴 = 𝐵 → ⟨𝐴, 𝐶⟩ = ⟨𝐵, 𝐶⟩)
3	1, 2	syl 14	1 ⊢ (𝜑 → ⟨𝐴, 𝐶⟩ = ⟨𝐵, 𝐶⟩)

Syntax hints:   → wi 4   = wceq 1402  ⟨cop 3711

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-un 3224  df-sn 3714  df-pr 3715  df-op 3717

This theorem is referenced by:  oteq1  3911  oteq2  3912  opth  4375  cbvoprab2  6155  djuf1olem  7387  dfplpq2  7715  ltexnqq  7769  nnanq0  7819  addpinq1  7825  prarloclemlo  7855  prarloclem3  7858  prarloclem5  7861  prsrriota  8149  caucvgsrlemfv  8152  caucvgsr  8163  pitonnlem2  8208  pitonn  8209  recidpirq  8219  ax1rid  8238  axrnegex  8240  nntopi  8255  axcaucvglemval  8258  fseq1m1p1  10485  frecuzrdglem  10831  frecuzrdgg  10836  frecuzrdgdomlem  10837  frecuzrdgfunlem  10839  frecuzrdgsuctlem  10843  pfxswrd  11461  swrdccat  11490  swrdccat3blem  11494  fsum2dlemstep  12184  fprod2dlemstep  12372  ennnfonelemp1  13280  ennnfonelemnn0  13296  setscomd  13376  imasaddvallemg  13619