Theorem opeq1d 3888

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 3882	. 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 3691

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 2214

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-v 2814  df-un 3214  df-sn 3694  df-pr 3695  df-op 3697

This theorem is referenced by:  oteq1  3891  oteq2  3892  opth  4352  cbvoprab2  6125  djuf1olem  7343  dfplpq2  7668  ltexnqq  7722  nnanq0  7772  addpinq1  7778  prarloclemlo  7808  prarloclem3  7811  prarloclem5  7814  prsrriota  8102  caucvgsrlemfv  8105  caucvgsr  8116  pitonnlem2  8161  pitonn  8162  recidpirq  8172  ax1rid  8191  axrnegex  8193  nntopi  8208  axcaucvglemval  8211  fseq1m1p1  10428  frecuzrdglem  10772  frecuzrdgg  10777  frecuzrdgdomlem  10778  frecuzrdgfunlem  10780  frecuzrdgsuctlem  10784  pfxswrd  11394  swrdccat  11423  swrdccat3blem  11427  fsum2dlemstep  12116  fprod2dlemstep  12304  ennnfonelemp1  13149  ennnfonelemnn0  13165  setscomd  13245  imasaddvallemg  13520