Theorem ralsd 17040

Description: Introduction rule for "all some" restricted to a class. This is the converse of rals1d 17043 and rals2d 17044 taken together. (Contributed by David A. Wheeler, 12-Jul-2026.)

Hypotheses

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

Assertion

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

Proof of Theorem ralsd

Step	Hyp	Ref	Expression
1		ralsd.1	. 2 |- ( ph -> A. x e. A ( ps -> ch ) )
2		ralsd.2	. 2 |- ( ph -> E. x e. A ps )
3		df-rals 17037	. 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 ) )

Syntax hints:   -> wi 4   A.wral 2528   E.wrex 2529   A.E.wrals 17035

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108

This theorem depends on definitions:  df-bi 117  df-rals 17037

This theorem is referenced by: (None)