Theorem rexanali

Description: A transformation of restricted quantifiers and logical connectives. (Contributed by NM, 4-Sep-2005) (Proof shortened by Wolf Lammen, 27-Dec-2019)

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

Proof

Step	Hyp	Ref	Expression
1		dfrex2	|- ( E. x e. A ( ph /\ -. ps ) <-> -. A. x e. A -. ( ph /\ -. ps ) )
2		iman	|- ( ( ph -> ps ) <-> -. ( ph /\ -. ps ) )
3	2	ralbii	|- ( A. x e. A ( ph -> ps ) <-> A. x e. A -. ( ph /\ -. ps ) )
4	1 3	xchbinxr	|- ( E. x e. A ( ph /\ -. ps ) <-> -. A. x e. A ( ph -> ps ) )