Theorem opeq1i 3807

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 3804	. 2 ⊢ (𝐴 = 𝐵 → ⟨𝐴, 𝐶⟩ = ⟨𝐵, 𝐶⟩)
3	1, 2	ax-mp 5	1 ⊢ ⟨𝐴, 𝐶⟩ = ⟨𝐵, 𝐶⟩

Syntax hints:   = wceq 1364  ⟨cop 3621

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-un 3157  df-sn 3624  df-pr 3625  df-op 3627

This theorem is referenced by:  caucvgsrlemfv  7851  caucvgsr  7862  pitonnlem1  7905  axi2m1  7935  axcaucvg  7960  ennnfonelem1  12564  2strstr1g  12739  2strop1g  12741