Theorem rexlimdvvva

Description: Inference from Theorem 19.23 of Margaris p. 90, for three restricted quantifiers. (Contributed by AV, 23-Aug-2025)

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

Proof

Step	Hyp	Ref	Expression
1		rexlimdvvva.1	|- ( ( ph /\ ( x e. A /\ y e. B /\ z e. C ) ) -> ( ps -> ch ) )
2		df-3an	|- ( ( x e. A /\ y e. B /\ z e. C ) <-> ( ( x e. A /\ y e. B ) /\ z e. C ) )
3	1	ex	|- ( ph -> ( ( x e. A /\ y e. B /\ z e. C ) -> ( ps -> ch ) ) )
4	2 3	biimtrrid	|- ( ph -> ( ( ( x e. A /\ y e. B ) /\ z e. C ) -> ( ps -> ch ) ) )
5	4	expdimp	|- ( ( ph /\ ( x e. A /\ y e. B ) ) -> ( z e. C -> ( ps -> ch ) ) )
6	5	rexlimdv	|- ( ( ph /\ ( x e. A /\ y e. B ) ) -> ( E. z e. C ps -> ch ) )
7	6	rexlimdvva	|- ( ph -> ( E. x e. A E. y e. B E. z e. C ps -> ch ) )