Theorem opeq1d 3783

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 3777	. 2 ⊢ (𝐴 = 𝐵 → ⟨𝐴, 𝐶⟩ = ⟨𝐵, 𝐶⟩)
3	1, 2	syl 14	1 ⊢ (𝜑 → ⟨𝐴, 𝐶⟩ = ⟨𝐵, 𝐶⟩)

Syntax hints:   → wi 4   = wceq 1353  ⟨cop 3595

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-un 3133  df-sn 3598  df-pr 3599  df-op 3601

This theorem is referenced by:  oteq1  3786  oteq2  3787  opth  4235  cbvoprab2  5943  djuf1olem  7047  dfplpq2  7348  ltexnqq  7402  nnanq0  7452  addpinq1  7458  prarloclemlo  7488  prarloclem3  7491  prarloclem5  7494  prsrriota  7782  caucvgsrlemfv  7785  caucvgsr  7796  pitonnlem2  7841  pitonn  7842  recidpirq  7852  ax1rid  7871  axrnegex  7873  nntopi  7888  axcaucvglemval  7891  fseq1m1p1  10088  frecuzrdglem  10404  frecuzrdgg  10409  frecuzrdgdomlem  10410  frecuzrdgfunlem  10412  frecuzrdgsuctlem  10416  fsum2dlemstep  11433  fprod2dlemstep  11621  ennnfonelemp1  12397  ennnfonelemnn0  12413  setscomd  12493