Theorem opelxpd

Description: Ordered pair membership in a Cartesian product, deduction form. (Contributed by Glauco Siliprandi, 3-Mar-2021)

	Ref	Expression
Hypotheses	opelxpd.1	$$⊢\phi \to A\in C$$
	opelxpd.2	$$⊢\phi \to B\in D$$
Assertion	opelxpd	$$⊢\phi \to ⟨A,B⟩\in \left(C\times D\right)$$

Proof

Step	Hyp	Ref	Expression
1		opelxpd.1	$$⊢\phi \to A\in C$$
2		opelxpd.2	$$⊢\phi \to B\in D$$
3		opelxpi	$$⊢\left(A\in C\wedge B\in D\right)\to ⟨A,B⟩\in \left(C\times D\right)$$
4	1 2 3	syl2anc	$$⊢\phi \to ⟨A,B⟩\in \left(C\times D\right)$$