Theorem opeq1d 3771

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 3765	. 2 ⊢ (𝐴 = 𝐵 → ⟨𝐴, 𝐶⟩ = ⟨𝐵, 𝐶⟩)
3	1, 2	syl 14	1 ⊢ (𝜑 → ⟨𝐴, 𝐶⟩ = ⟨𝐵, 𝐶⟩)

Syntax hints:   → wi 4   = wceq 1348  ⟨cop 3586

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-un 3125  df-sn 3589  df-pr 3590  df-op 3592

This theorem is referenced by:  oteq1  3774  oteq2  3775  opth  4222  cbvoprab2  5926  djuf1olem  7030  dfplpq2  7316  ltexnqq  7370  nnanq0  7420  addpinq1  7426  prarloclemlo  7456  prarloclem3  7459  prarloclem5  7462  prsrriota  7750  caucvgsrlemfv  7753  caucvgsr  7764  pitonnlem2  7809  pitonn  7810  recidpirq  7820  ax1rid  7839  axrnegex  7841  nntopi  7856  axcaucvglemval  7859  fseq1m1p1  10051  frecuzrdglem  10367  frecuzrdgg  10372  frecuzrdgdomlem  10373  frecuzrdgfunlem  10375  frecuzrdgsuctlem  10379  fsum2dlemstep  11397  fprod2dlemstep  11585  ennnfonelemp1  12361  ennnfonelemnn0  12377