Theorem opeq12i 3861

Description: Equality inference for ordered pairs. (Contributed by NM, 16-Dec-2006.) (Proof shortened by Eric Schmidt, 4-Apr-2007.)

Hypotheses

Ref	Expression
opeq1i.1	⊢ 𝐴 = 𝐵
opeq12i.2	⊢ 𝐶 = 𝐷

Assertion

Ref	Expression
opeq12i	⊢ ⟨𝐴, 𝐶⟩ = ⟨𝐵, 𝐷⟩

Proof of Theorem opeq12i

Step	Hyp	Ref	Expression
1		opeq1i.1	. 2 ⊢ 𝐴 = 𝐵
2		opeq12i.2	. 2 ⊢ 𝐶 = 𝐷
3		opeq12 3858	. 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → ⟨𝐴, 𝐶⟩ = ⟨𝐵, 𝐷⟩)
4	1, 2, 3	mp2an 426	1 ⊢ ⟨𝐴, 𝐶⟩ = ⟨𝐵, 𝐷⟩

Syntax hints:   = wceq 1395  ⟨cop 3669

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-un 3201  df-sn 3672  df-pr 3673  df-op 3675

This theorem is referenced by:  addpinq1  7647  genipv  7692  ltexpri  7796  recexpr  7821  cauappcvgprlemladdru  7839  cauappcvgprlemladdrl  7840  cauappcvgpr  7845  caucvgprlemcl  7859  caucvgprlemladdrl  7861  caucvgpr  7865  caucvgprprlemval  7871  caucvgprprlemnbj  7876  caucvgprprlemmu  7878  caucvgprprlemclphr  7888  caucvgprprlemaddq  7891  caucvgprprlem1  7892  caucvgprprlem2  7893  caucvgsr  7985  pitonnlem1  8028  axi2m1  8058  axcaucvg  8083