Theorem opeq1d 3711

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 3705	. 2 ⊢ (𝐴 = 𝐵 → ⟨𝐴, 𝐶⟩ = ⟨𝐵, 𝐶⟩)
3	1, 2	syl 14	1 ⊢ (𝜑 → ⟨𝐴, 𝐶⟩ = ⟨𝐵, 𝐶⟩)

Syntax hints:   → wi 4   = wceq 1331  ⟨cop 3530

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-un 3075  df-sn 3533  df-pr 3534  df-op 3536

This theorem is referenced by:  oteq1  3714  oteq2  3715  opth  4159  cbvoprab2  5844  djuf1olem  6938  dfplpq2  7162  ltexnqq  7216  nnanq0  7266  addpinq1  7272  prarloclemlo  7302  prarloclem3  7305  prarloclem5  7308  prsrriota  7596  caucvgsrlemfv  7599  caucvgsr  7610  pitonnlem2  7655  pitonn  7656  recidpirq  7666  ax1rid  7685  axrnegex  7687  nntopi  7702  axcaucvglemval  7705  fseq1m1p1  9875  frecuzrdglem  10184  frecuzrdgg  10189  frecuzrdgdomlem  10190  frecuzrdgfunlem  10192  frecuzrdgsuctlem  10196  fsum2dlemstep  11203  ennnfonelemp1  11919  ennnfonelemnn0  11935