Theorem opeq1i 3655

Description: Equality inference for ordered pairs. (Contributed by NM, 16-Dec-2006.)

Hypothesis

Ref	Expression
opeq1i.1	⊢ 𝐴 = 𝐵

Assertion

Ref	Expression
opeq1i	⊢ ⟨𝐴, 𝐶⟩ = ⟨𝐵, 𝐶⟩

Proof of Theorem opeq1i

Step	Hyp	Ref	Expression
1		opeq1i.1	. 2 ⊢ 𝐴 = 𝐵
2		opeq1 3652	. 2 ⊢ (𝐴 = 𝐵 → ⟨𝐴, 𝐶⟩ = ⟨𝐵, 𝐶⟩)
3	1, 2	ax-mp 7	1 ⊢ ⟨𝐴, 𝐶⟩ = ⟨𝐵, 𝐶⟩

Syntax hints:   = wceq 1299  ⟨cop 3477

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082

This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-v 2643  df-un 3025  df-sn 3480  df-pr 3481  df-op 3483

This theorem is referenced by:  caucvgsrlemfv  7486  caucvgsr  7497  pitonnlem1  7532  axi2m1  7560  axcaucvg  7585  ennnfonelem1  11712  2strstr1g  11844  2strop1g  11846