Theorem rals2d

Description: Deduction rule: Given "all some" applied to a class, you can extract the "there exists" part. Note that the witness must satisfy the antecedent ps , not merely be a member of A . (Contributed by David A. Wheeler, 20-Oct-2018) (Revised by David A. Wheeler, 12-Jul-2026)

	Ref	Expression
Hypothesis	rals2d.1	|- ( ph -> AE x e. A ( ps -> ch ) )
Assertion	rals2d	|- ( ph -> E. x e. A ps )

Proof

Step	Hyp	Ref	Expression
1		rals2d.1	|- ( ph -> AE x e. A ( ps -> ch ) )
2		df-rals	|- ( AE x e. A ( ps -> ch ) <-> ( A. x e. A ( ps -> ch ) /\ E. x e. A ps ) )
3	1 2	sylib	|- ( ph -> ( A. x e. A ( ps -> ch ) /\ E. x e. A ps ) )
4	3	simprd	|- ( ph -> E. x e. A ps )