Theorem exanali

Description: A transformation of quantifiers and logical connectives. (Contributed by NM, 25-Mar-1996) (Proof shortened by Wolf Lammen, 4-Sep-2014)

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

Proof

Step	Hyp	Ref	Expression
1		annim	|- ( ( ph /\ -. ps ) <-> -. ( ph -> ps ) )
2	1	exbii	|- ( E. x ( ph /\ -. ps ) <-> E. x -. ( ph -> ps ) )
3		exnal	|- ( E. x -. ( ph -> ps ) <-> -. A. x ( ph -> ps ) )
4	2 3	bitri	|- ( E. x ( ph /\ -. ps ) <-> -. A. x ( ph -> ps ) )