Theorem ceqsexv

Description: Elimination of an existential quantifier, using implicit substitution. (Contributed by NM, 2-Mar-1995) Avoid ax-12 . (Revised by Gino Giotto, 12-Oct-2024) (Proof shortened by Wolf Lammen, 22-Jan-2025)

	Ref	Expression
Hypotheses	ceqsexv.1	|- A e. _V
	ceqsexv.2	|- ( x = A -> ( ph <-> ps ) )
Assertion	ceqsexv	|- ( E. x ( x = A /\ ph ) <-> ps )

Proof

Step	Hyp	Ref	Expression
1		ceqsexv.1	|- A e. _V
2		ceqsexv.2	|- ( x = A -> ( ph <-> ps ) )
3		alinexa	|- ( A. x ( x = A -> -. ph ) <-> -. E. x ( x = A /\ ph ) )
4	2	notbid	|- ( x = A -> ( -. ph <-> -. ps ) )
5	1 4	ceqsalv	|- ( A. x ( x = A -> -. ph ) <-> -. ps )
6	3 5	bitr3i	|- ( -. E. x ( x = A /\ ph ) <-> -. ps )
7	6	con4bii	|- ( E. x ( x = A /\ ph ) <-> ps )