Theorem reurexprg

Description: Convert a restricted existential uniqueness over a pair to a restricted existential quantification and an implication . (Contributed by AV, 3-Apr-2023)

	Ref	Expression
Hypotheses	reuprg.1	|- ( x = A -> ( ph <-> ps ) )
	reuprg.2	|- ( x = B -> ( ph <-> ch ) )
Assertion	reurexprg	|- ( ( A e. V /\ B e. W ) -> ( E! x e. { A , B } ph <-> ( E. x e. { A , B } ph /\ ( ( ch /\ ps ) -> A = B ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		reuprg.1	|- ( x = A -> ( ph <-> ps ) )
2		reuprg.2	|- ( x = B -> ( ph <-> ch ) )
3	1 2	reuprg	|- ( ( A e. V /\ B e. W ) -> ( E! x e. { A , B } ph <-> ( ( ps \/ ch ) /\ ( ( ch /\ ps ) -> A = B ) ) ) )
4	1 2	rexprg	|- ( ( A e. V /\ B e. W ) -> ( E. x e. { A , B } ph <-> ( ps \/ ch ) ) )
5	4	bicomd	|- ( ( A e. V /\ B e. W ) -> ( ( ps \/ ch ) <-> E. x e. { A , B } ph ) )
6	5	anbi1d	|- ( ( A e. V /\ B e. W ) -> ( ( ( ps \/ ch ) /\ ( ( ch /\ ps ) -> A = B ) ) <-> ( E. x e. { A , B } ph /\ ( ( ch /\ ps ) -> A = B ) ) ) )
7	3 6	bitrd	|- ( ( A e. V /\ B e. W ) -> ( E! x e. { A , B } ph <-> ( E. x e. { A , B } ph /\ ( ( ch /\ ps ) -> A = B ) ) ) )