Theorem 2rspcedvdw

Description: Double application of rspcedvdw . (Contributed by SN, 24-Aug-2024)

	Ref	Expression
Hypotheses	2rspcedvdw.1	$$⊢x=A\to \left(\psi \leftrightarrow \chi \right)$$
	2rspcedvdw.2	$$⊢y=B\to \left(\chi \leftrightarrow \theta \right)$$
	2rspcedvdw.a	$$⊢\phi \to A\in X$$
	2rspcedvdw.b	$$⊢\phi \to B\in Y$$
	2rspcedvdw.3	$$⊢\phi \to \theta$$
Assertion	2rspcedvdw	$$⊢\phi \to \exists x\in X\exists y\in Y\psi$$

Proof

Step	Hyp	Ref	Expression
1		2rspcedvdw.1	$$⊢x=A\to \left(\psi \leftrightarrow \chi \right)$$
2		2rspcedvdw.2	$$⊢y=B\to \left(\chi \leftrightarrow \theta \right)$$
3		2rspcedvdw.a	$$⊢\phi \to A\in X$$
4		2rspcedvdw.b	$$⊢\phi \to B\in Y$$
5		2rspcedvdw.3	$$⊢\phi \to \theta$$
6	1 2	rspc2ev	$$⊢\left(A\in X\wedge B\in Y\wedge \theta \right)\to \exists x\in X\exists y\in Y\psi$$
7	3 4 5 6	syl3anc	$$⊢\phi \to \exists x\in X\exists y\in Y\psi$$