Theorem r19.36zv

Description: Restricted quantifier version of Theorem 19.36 of Margaris p. 90. It is valid only when the domain of quantification is not empty. (Contributed by NM, 20-Sep-2003)

	Ref	Expression
Assertion	r19.36zv	|- ( A =/= (/) -> ( E. x e. A ( ph -> ps ) <-> ( A. x e. A ph -> ps ) ) )

Proof

Step	Hyp	Ref	Expression
1		r19.35	|- ( E. x e. A ( ph -> ps ) <-> ( A. x e. A ph -> E. x e. A ps ) )
2		r19.9rzv	|- ( A =/= (/) -> ( ps <-> E. x e. A ps ) )
3	2	imbi2d	|- ( A =/= (/) -> ( ( A. x e. A ph -> ps ) <-> ( A. x e. A ph -> E. x e. A ps ) ) )
4	1 3	bitr4id	|- ( A =/= (/) -> ( E. x e. A ( ph -> ps ) <-> ( A. x e. A ph -> ps ) ) )