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Metamath Proof Explorer

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		r19.26	\|- ( A. x e. A ( ph /\ ps ) <-> ( A. x e. A ph /\ A. x e. A ps ) )
3	1	r19.3rz	\|- ( A =/= (/) -> ( ph <-> A. x e. A ph ) )
4	3	anbi1d	\|- ( A =/= (/) -> ( ( ph /\ A. x e. A ps ) <-> ( A. x e. A ph /\ A. x e. A ps ) ) )
5	2 4	bitr4id	\|- ( A =/= (/) -> ( A. x e. A ( ph /\ ps ) <-> ( ph /\ A. x e. A ps ) ) )