Theorem r19.28z

Description: Restricted quantifier version of Theorem 19.28 of Margaris p. 90. It is valid only when the domain of quantification is not empty. (Contributed by NM, 26-Oct-2010)

	Ref	Expression
Hypothesis	r19.3rz.1	|- F/ x ph
Assertion	r19.28z	|- ( A =/= (/) -> ( A. x e. A ( ph /\ ps ) <-> ( ph /\ A. x e. A ps ) ) )

Proof

Step	Hyp	Ref	Expression
1		r19.3rz.1	|- F/ x ph
2	1	r19.3rz	|- ( A =/= (/) -> ( ph <-> A. x e. A ph ) )
3	2	anbi1d	|- ( A =/= (/) -> ( ( ph /\ A. x e. A ps ) <-> ( A. x e. A ph /\ A. x e. A ps ) ) )
4		r19.26	|- ( A. x e. A ( ph /\ ps ) <-> ( A. x e. A ph /\ A. x e. A ps ) )
5	3 4	syl6rbbr	|- ( A =/= (/) -> ( A. x e. A ( ph /\ ps ) <-> ( ph /\ A. x e. A ps ) ) )