Theorem opelopabd

Description: Membership of an ordere pair in a class abstraction of ordered pairs. (Contributed by BJ, 17-Dec-2023)

	Ref	Expression
Hypotheses	opelopabd.xph	$$⊢\phi \to \forall x\phi$$
	opelopabd.yph	$$⊢\phi \to \forall y\phi$$
	opelopabd.xch	$$⊢\phi \to Ⅎx\chi$$
	opelopabd.ych	$$⊢\phi \to Ⅎy\chi$$
	opelopabd.exa	$$⊢\phi \to A\in U$$
	opelopabd.exb	$$⊢\phi \to B\in V$$
	opelopabd.is	$$⊢\left(\phi \wedge \left(x=A\wedge y=B\right)\right)\to \left(\psi \leftrightarrow \chi \right)$$
Assertion	opelopabd	$$⊢\phi \to \left(⟨A,B⟩\in \left\{⟨x,y⟩|\psi \right\}\leftrightarrow \chi \right)$$

Proof

Step	Hyp	Ref	Expression
1		opelopabd.xph	$$⊢\phi \to \forall x\phi$$
2		opelopabd.yph	$$⊢\phi \to \forall y\phi$$
3		opelopabd.xch	$$⊢\phi \to Ⅎx\chi$$
4		opelopabd.ych	$$⊢\phi \to Ⅎy\chi$$
5		opelopabd.exa	$$⊢\phi \to A\in U$$
6		opelopabd.exb	$$⊢\phi \to B\in V$$
7		opelopabd.is	$$⊢\left(\phi \wedge \left(x=A\wedge y=B\right)\right)\to \left(\psi \leftrightarrow \chi \right)$$
8		elopab	$$⊢⟨A,B⟩\in \left\{⟨x,y⟩|\psi \right\}\leftrightarrow \exists x\exists y\left(⟨A,B⟩=⟨x,y⟩\wedge \psi \right)$$
9	1 2 3 4 5 6 7	copsex2d	$$⊢\phi \to \left(\exists x\exists y\left(⟨A,B⟩=⟨x,y⟩\wedge \psi \right)\leftrightarrow \chi \right)$$
10	8 9	syl5bb	$$⊢\phi \to \left(⟨A,B⟩\in \left\{⟨x,y⟩|\psi \right\}\leftrightarrow \chi \right)$$