Theorem ralseud 17180

Description: Introduction rule for "all some one" restricted to a class. This is the converse of ralseu1d 17183 and ralseu2d 17184 taken together. (Contributed by David A. Wheeler, 22-Jul-2026.)

Hypotheses

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

Assertion

Ref	Expression
ralseud	|- ( ph -> A.E! x e. A ( ps -> ch ) )

Proof of Theorem ralseud

Step	Hyp	Ref	Expression
1		ralseud.1	. 2 |- ( ph -> A. x e. A ( ps -> ch ) )
2		ralseud.2	. 2 |- ( ph -> E! x e. A ps )
3		df-ralseu 17175	. 2 |- ( A.E! x e. A ( ps -> ch ) <-> ( A. x e. A ( ps -> ch ) /\ E! x e. A ps ) )
4	1, 2, 3	sylanbrc 421	1 |- ( ph -> A.E! x e. A ( ps -> ch ) )

This proof depends on syntax axioms:   -> wi 4   A.wral 2528   E!wreu 2530   A.E!wralseu 17173

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108

This proof depends on definitions:  df-bi 117  df-ralseu 17175

This theorem is used by: (None)