Theorem r19.27z

Description: Restricted quantifier version of Theorem 19.27 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	r19.27z.1	|- F/ x ps
Assertion	r19.27z	|- ( A =/= (/) -> ( A. x e. A ( ph /\ ps ) <-> ( A. x e. A ph /\ ps ) ) )

Proof

Step	Hyp	Ref	Expression
1		r19.27z.1	|- F/ x ps
2		r19.26	|- ( A. x e. A ( ph /\ ps ) <-> ( A. x e. A ph /\ A. x e. A ps ) )
3	1	r19.3rz	|- ( A =/= (/) -> ( ps <-> A. x e. A ps ) )
4	3	anbi2d	|- ( A =/= (/) -> ( ( A. x e. A ph /\ ps ) <-> ( A. x e. A ph /\ A. x e. A ps ) ) )
5	2 4	bitr4id	|- ( A =/= (/) -> ( A. x e. A ( ph /\ ps ) <-> ( A. x e. A ph /\ ps ) ) )