Theorem rexlimdv3d

Description: An extended version of rexlimdvv to include three set variables. (Contributed by Igor Ieskov, 21-Jan-2024)

	Ref	Expression
Hypothesis	rexlimdv3d.1	|- ( ph -> ( ( x e. A /\ y e. B /\ z e. C ) -> ( ps -> ch ) ) )
Assertion	rexlimdv3d	|- ( ph -> ( E. x e. A E. y e. B E. z e. C ps -> ch ) )

Proof

Step	Hyp	Ref	Expression
1		rexlimdv3d.1	|- ( ph -> ( ( x e. A /\ y e. B /\ z e. C ) -> ( ps -> ch ) ) )
2	1	3expd	|- ( ph -> ( x e. A -> ( y e. B -> ( z e. C -> ( ps -> ch ) ) ) ) )
3	2	imp4d	|- ( ph -> ( ( x e. A /\ ( y e. B /\ z e. C ) ) -> ( ps -> ch ) ) )
4	3	expdimp	|- ( ( ph /\ x e. A ) -> ( ( y e. B /\ z e. C ) -> ( ps -> ch ) ) )
5	4	rexlimdvv	|- ( ( ph /\ x e. A ) -> ( E. y e. B E. z e. C ps -> ch ) )
6	5	rexlimdva	|- ( ph -> ( E. x e. A E. y e. B E. z e. C ps -> ch ) )