Theorem 2rspcedvdw

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

	Ref	Expression
Hypotheses	2rspcedvdw.1	|- ( x = A -> ( ps <-> ch ) )
	2rspcedvdw.2	|- ( y = B -> ( ch <-> th ) )
	2rspcedvdw.a	|- ( ph -> A e. X )
	2rspcedvdw.b	|- ( ph -> B e. Y )
	2rspcedvdw.3	|- ( ph -> th )
Assertion	2rspcedvdw	|- ( ph -> E. x e. X E. y e. Y ps )

Proof

Step	Hyp	Ref	Expression
1		2rspcedvdw.1	|- ( x = A -> ( ps <-> ch ) )
2		2rspcedvdw.2	|- ( y = B -> ( ch <-> th ) )
3		2rspcedvdw.a	|- ( ph -> A e. X )
4		2rspcedvdw.b	|- ( ph -> B e. Y )
5		2rspcedvdw.3	|- ( ph -> th )
6	1 2	rspc2ev	|- ( ( A e. X /\ B e. Y /\ th ) -> E. x e. X E. y e. Y ps )
7	3 4 5 6	syl3anc	|- ( ph -> E. x e. X E. y e. Y ps )