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Metamath Proof Explorer

Theorem r19.3rzv

Description: Restricted quantification of wff not containing quantified variable. (Contributed by NM, 10-Mar-1997) Avoid ax-12 . (Revised by TM, 16-Feb-2026)

		Ref	Expression
	Assertion	r19.3rzv	\|- ( A =/= (/) -> ( ph <-> A. x e. A ph ) )

Proof

Step	Hyp	Ref	Expression
1		ax-1	\|- ( ph -> ( x e. A -> ph ) )
2	1	ralrimiv	\|- ( ph -> A. x e. A ph )
3		rspn0	\|- ( A =/= (/) -> ( A. x e. A ph -> ph ) )
4	2 3	impbid2	\|- ( A =/= (/) -> ( ph <-> A. x e. A ph ) )