Theorem opeq1d 3894

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 3888	. 2 ⊢ (𝐴 = 𝐵 → ⟨𝐴, 𝐶⟩ = ⟨𝐵, 𝐶⟩)
3	1, 2	syl 14	1 ⊢ (𝜑 → ⟨𝐴, 𝐶⟩ = ⟨𝐵, 𝐶⟩)

Syntax hints:   → wi 4   = wceq 1398  ⟨cop 3697

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-v 2817  df-un 3218  df-sn 3700  df-pr 3701  df-op 3703

This theorem is referenced by:  oteq1  3897  oteq2  3898  opth  4358  cbvoprab2  6134  djuf1olem  7357  dfplpq2  7685  ltexnqq  7739  nnanq0  7789  addpinq1  7795  prarloclemlo  7825  prarloclem3  7828  prarloclem5  7831  prsrriota  8119  caucvgsrlemfv  8122  caucvgsr  8133  pitonnlem2  8178  pitonn  8179  recidpirq  8189  ax1rid  8208  axrnegex  8210  nntopi  8225  axcaucvglemval  8228  fseq1m1p1  10451  frecuzrdglem  10797  frecuzrdgg  10802  frecuzrdgdomlem  10803  frecuzrdgfunlem  10805  frecuzrdgsuctlem  10809  pfxswrd  11423  swrdccat  11452  swrdccat3blem  11456  fsum2dlemstep  12145  fprod2dlemstep  12333  ennnfonelemp1  13241  ennnfonelemnn0  13257  setscomd  13337  imasaddvallemg  13612