Theorem r19.29

Description: Restricted quantifier version of 19.29 . See also r19.29r . (Contributed by NM, 31-Aug-1999) (Proof shortened by Andrew Salmon, 30-May-2011)

	Ref	Expression
Assertion	r19.29	|- ( ( A. x e. A ph /\ E. x e. A ps ) -> E. x e. A ( ph /\ ps ) )

Proof

Step	Hyp	Ref	Expression
1		pm3.2	|- ( ph -> ( ps -> ( ph /\ ps ) ) )
2	1	ralimi	|- ( A. x e. A ph -> A. x e. A ( ps -> ( ph /\ ps ) ) )
3		rexim	|- ( A. x e. A ( ps -> ( ph /\ ps ) ) -> ( E. x e. A ps -> E. x e. A ( ph /\ ps ) ) )
4	2 3	syl	|- ( A. x e. A ph -> ( E. x e. A ps -> E. x e. A ( ph /\ ps ) ) )
5	4	imp	|- ( ( A. x e. A ph /\ E. x e. A ps ) -> E. x e. A ( ph /\ ps ) )