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Metamath Proof Explorer

Theorem opeq12i

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

Ref Expression
Hypotheses opeq1i.1 ⊢ 𝐴 = 𝐵
opeq12i.2 ⊢ 𝐶 = 𝐷
Assertion opeq12i ⟨ 𝐴 , 𝐶 ⟩ = ⟨ 𝐵 , 𝐷 ⟩

Proof

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