Theorem opeq12i 3814

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 3811	. 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → ⟨𝐴, 𝐶⟩ = ⟨𝐵, 𝐷⟩)
4	1, 2, 3	mp2an 426	1 ⊢ ⟨𝐴, 𝐶⟩ = ⟨𝐵, 𝐷⟩

Syntax hints:   = wceq 1364  ⟨cop 3626

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-un 3161  df-sn 3629  df-pr 3630  df-op 3632

This theorem is referenced by:  addpinq1  7550  genipv  7595  ltexpri  7699  recexpr  7724  cauappcvgprlemladdru  7742  cauappcvgprlemladdrl  7743  cauappcvgpr  7748  caucvgprlemcl  7762  caucvgprlemladdrl  7764  caucvgpr  7768  caucvgprprlemval  7774  caucvgprprlemnbj  7779  caucvgprprlemmu  7781  caucvgprprlemclphr  7791  caucvgprprlemaddq  7794  caucvgprprlem1  7795  caucvgprprlem2  7796  caucvgsr  7888  pitonnlem1  7931  axi2m1  7961  axcaucvg  7986