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Metamath Proof Explorer

Theorem rr19.3v

Description: Restricted quantifier version of Theorem 19.3 of Margaris p. 89. We don't need the nonempty class condition of r19.3rzv when there is an outer quantifier. (Contributed by NM, 25-Oct-2012)

		Ref	Expression
	Assertion	rr19.3v	\|- ( A. x e. A A. y e. A ph <-> A. x e. A ph )

Proof

Step	Hyp	Ref	Expression
1		biidd	\|- ( y = x -> ( ph <-> ph ) )
2	1	rspcv	\|- ( x e. A -> ( A. y e. A ph -> ph ) )
3	2	ralimia	\|- ( A. x e. A A. y e. A ph -> A. x e. A ph )
4		ax-1	\|- ( ph -> ( y e. A -> ph ) )
5	4	ralrimiv	\|- ( ph -> A. y e. A ph )
6	5	ralimi	\|- ( A. x e. A ph -> A. x e. A A. y e. A ph )
7	3 6	impbii	\|- ( A. x e. A A. y e. A ph <-> A. x e. A ph )