Theorem brabd

Description: Expressing that two sets are related by a binary relation which is expressed as a class abstraction of ordered pairs. (Contributed by BJ, 17-Dec-2023)

	Ref	Expression
Hypotheses	brabd.exa	$$⊢\phi \to A\in U$$
	brabd.exb	$$⊢\phi \to B\in V$$
	brabd.def	$$⊢\phi \to R=\left\{⟨x,y⟩|\psi \right\}$$
	brabd.is	$$⊢\left(\phi \wedge \left(x=A\wedge y=B\right)\right)\to \left(\psi \leftrightarrow \chi \right)$$
Assertion	brabd	$$⊢\phi \to \left(ARB\leftrightarrow \chi \right)$$

Proof

Step	Hyp	Ref	Expression
1		brabd.exa	$$⊢\phi \to A\in U$$
2		brabd.exb	$$⊢\phi \to B\in V$$
3		brabd.def	$$⊢\phi \to R=\left\{⟨x,y⟩|\psi \right\}$$
4		brabd.is	$$⊢\left(\phi \wedge \left(x=A\wedge y=B\right)\right)\to \left(\psi \leftrightarrow \chi \right)$$
5		ax-5	$$⊢\phi \to \forall x\phi$$
6		ax-5	$$⊢\phi \to \forall y\phi$$
7		nfvd	$$⊢\phi \to Ⅎx\chi$$
8		nfvd	$$⊢\phi \to Ⅎy\chi$$
9	5 6 7 8 1 2 3 4	brabd0	$$⊢\phi \to \left(ARB\leftrightarrow \chi \right)$$