Theorem opeq12i 3909

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 3906	. 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → ⟨𝐴, 𝐶⟩ = ⟨𝐵, 𝐷⟩)
4	1, 2, 3	mp2an 430	1 ⊢ ⟨𝐴, 𝐶⟩ = ⟨𝐵, 𝐷⟩

This proof depends on syntax axioms:   = wceq 1402  ⟨cop 3712

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-un 3224  df-sn 3715  df-pr 3716  df-op 3718

This theorem is used by:  addpinq1  7831  genipv  7876  ltexpri  7980  recexpr  8005  cauappcvgprlemladdru  8023  cauappcvgprlemladdrl  8024  cauappcvgpr  8029  caucvgprlemcl  8043  caucvgprlemladdrl  8045  caucvgpr  8049  caucvgprprlemval  8055  caucvgprprlemnbj  8060  caucvgprprlemmu  8062  caucvgprprlemclphr  8072  caucvgprprlemaddq  8075  caucvgprprlem1  8076  caucvgprprlem2  8077  caucvgsr  8169  pitonnlem1  8212  axi2m1  8242  axcaucvg  8267  konigsbergvtx  16723  konigsbergiedg  16724