Theorem raln

Description: Restricted universally quantified negation expressed as a universally quantified negation. (Contributed by BJ, 16-Jul-2021)

	Ref	Expression
Assertion	raln	|- ( A. x e. A -. ph <-> A. x -. ( x e. A /\ ph ) )

Proof

Step	Hyp	Ref	Expression
1		df-ral	|- ( A. x e. A -. ph <-> A. x ( x e. A -> -. ph ) )
2		imnang	|- ( A. x ( x e. A -> -. ph ) <-> A. x -. ( x e. A /\ ph ) )
3	1 2	bitri	|- ( A. x e. A -. ph <-> A. x -. ( x e. A /\ ph ) )