Theorem expandrexn

Description: Expand a restricted existential quantifier to primitives while contracting a double negation. (Contributed by Rohan Ridenour, 13-Aug-2023)

	Ref	Expression
Hypothesis	expandrexn.1	|- ( ph <-> -. ps )
Assertion	expandrexn	|- ( E. x e. A ph <-> -. A. x ( x e. A -> ps ) )

Proof

Step	Hyp	Ref	Expression
1		expandrexn.1	|- ( ph <-> -. ps )
2	1	rexbii	|- ( E. x e. A ph <-> E. x e. A -. ps )
3		df-rex	|- ( E. x e. A -. ps <-> E. x ( x e. A /\ -. ps ) )
4		exanali	|- ( E. x ( x e. A /\ -. ps ) <-> -. A. x ( x e. A -> ps ) )
5	2 3 4	3bitri	|- ( E. x e. A ph <-> -. A. x ( x e. A -> ps ) )