Theorem opeq12d

Description: Equality deduction for ordered pairs. (Contributed by NM, 16-Dec-2006) (Proof shortened by Andrew Salmon, 29-Jun-2011)

	Ref	Expression
Hypotheses	opeq1d.1	$$⊢\phi \to A=B$$
	opeq12d.2	$$⊢\phi \to C=D$$
Assertion	opeq12d	$$⊢\phi \to ⟨A,C⟩=⟨B,D⟩$$

Proof

Step	Hyp	Ref	Expression
1		opeq1d.1	$$⊢\phi \to A=B$$
2		opeq12d.2	$$⊢\phi \to C=D$$
3		opeq12	$$⊢\left(A=B\wedge C=D\right)\to ⟨A,C⟩=⟨B,D⟩$$
4	1 2 3	syl2anc	$$⊢\phi \to ⟨A,C⟩=⟨B,D⟩$$