Theorem ralseud

Description: Introduction rule for "all some one" restricted to a class. This is the converse of ralseu1d and ralseu2d taken together. (Contributed by David A. Wheeler, 21-Jul-2026)

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

Proof

Step	Hyp	Ref	Expression
1		ralseud.1	|- ( ph -> A. x e. A ( ps -> ch ) )
2		ralseud.2	|- ( ph -> E! x e. A ps )
3		df-ralseu	|- ( 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 ) )