Theorem opeq1d 3786

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 3780	. 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 3597

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 2741  df-un 3135  df-sn 3600  df-pr 3601  df-op 3603

This theorem is referenced by:  oteq1  3789  oteq2  3790  opth  4239  cbvoprab2  5950  djuf1olem  7054  dfplpq2  7355  ltexnqq  7409  nnanq0  7459  addpinq1  7465  prarloclemlo  7495  prarloclem3  7498  prarloclem5  7501  prsrriota  7789  caucvgsrlemfv  7792  caucvgsr  7803  pitonnlem2  7848  pitonn  7849  recidpirq  7859  ax1rid  7878  axrnegex  7880  nntopi  7895  axcaucvglemval  7898  fseq1m1p1  10097  frecuzrdglem  10413  frecuzrdgg  10418  frecuzrdgdomlem  10419  frecuzrdgfunlem  10421  frecuzrdgsuctlem  10425  fsum2dlemstep  11444  fprod2dlemstep  11632  ennnfonelemp1  12409  ennnfonelemnn0  12425  setscomd  12505  imasaddvallemg  12741