Theorem r19.37zv

Description: Restricted quantifier version of Theorem 19.37 of Margaris p. 90. It is valid only when the domain of quantification is not empty. (Contributed by Paul Chapman, 8-Oct-2007)

	Ref	Expression
Assertion	r19.37zv	|- ( A =/= (/) -> ( E. x e. A ( ph -> ps ) <-> ( ph -> E. x e. A ps ) ) )

Proof

Step	Hyp	Ref	Expression
1		r19.3rzv	|- ( A =/= (/) -> ( ph <-> A. x e. A ph ) )
2	1	imbi1d	|- ( A =/= (/) -> ( ( ph -> E. x e. A ps ) <-> ( A. x e. A ph -> E. x e. A ps ) ) )
3		r19.35	|- ( E. x e. A ( ph -> ps ) <-> ( A. x e. A ph -> E. x e. A ps ) )
4	2 3	syl6rbbr	|- ( A =/= (/) -> ( E. x e. A ( ph -> ps ) <-> ( ph -> E. x e. A ps ) ) )