Theorem opeq1i 3870

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 3867	. 2 ⊢ (𝐴 = 𝐵 → ⟨𝐴, 𝐶⟩ = ⟨𝐵, 𝐶⟩)
3	1, 2	ax-mp 5	1 ⊢ ⟨𝐴, 𝐶⟩ = ⟨𝐵, 𝐶⟩

Syntax hints:   = wceq 1398  ⟨cop 3676

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805  df-un 3205  df-sn 3679  df-pr 3680  df-op 3682

This theorem is referenced by:  caucvgsrlemfv  8054  caucvgsr  8065  pitonnlem1  8108  axi2m1  8138  axcaucvg  8163  ennnfonelem1  13089  2strstr1g  13266  2strop1g  13268  setsiedg  15973