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Metamath Proof Explorer

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		r19.35	\|- ( E. x e. A ( ph -> ps ) <-> ( A. x e. A ph -> E. x e. A ps ) )
2		r19.3rzv	\|- ( A =/= (/) -> ( ph <-> A. x e. A ph ) )
3	2	imbi1d	\|- ( A =/= (/) -> ( ( ph -> E. x e. A ps ) <-> ( A. x e. A ph -> E. x e. A ps ) ) )
4	1 3	bitr4id	\|- ( A =/= (/) -> ( E. x e. A ( ph -> ps ) <-> ( ph -> E. x e. A ps ) ) )