Theorem bj-cbvex

Description: Changing a bound variable (existential quantification case) in a weak axiomatization, assuming that all variables denote (which is valid in inclusive free logic) and that equality is symmetric. (Contributed by BJ, 12-Mar-2023) (Proof modification is discouraged.)

	Ref	Expression
Hypotheses	bj-cbval.denote	|- A. y E. x x = y
	bj-cbval.denote2	|- A. x E. y y = x
	bj-cbval.maj	|- ( x = y -> ( ph <-> ps ) )
	bj-cbval.equcomiv	|- ( y = x -> x = y )
Assertion	bj-cbvex	|- ( E. x ph <-> E. y ps )

Proof

Step	Hyp	Ref	Expression
1		bj-cbval.denote	|- A. y E. x x = y
2		bj-cbval.denote2	|- A. x E. y y = x
3		bj-cbval.maj	|- ( x = y -> ( ph <-> ps ) )
4		bj-cbval.equcomiv	|- ( y = x -> x = y )
5	3	biimpd	|- ( x = y -> ( ph -> ps ) )
6	4 5	syl	|- ( y = x -> ( ph -> ps ) )
7	6 2	bj-cbveximi	|- ( E. x ph -> E. y ps )
8	3	biimprd	|- ( x = y -> ( ps -> ph ) )
9	8 1	bj-cbveximi	|- ( E. y ps -> E. x ph )
10	7 9	impbii	|- ( E. x ph <-> E. y ps )