Theorem ralsd

Description: Introduction rule for "all some" restricted to a class. This is the converse of rals1d and rals2d taken together. (Contributed by David A. Wheeler, 12-Jul-2026)

	Ref	Expression
Hypotheses	ralsd.1	|- ( ph -> A. x e. A ( ps -> ch ) )
	ralsd.2	|- ( ph -> E. x e. A ps )
Assertion	ralsd	|- ( ph -> AE x e. A ( ps -> ch ) )

Proof

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