Theorem rexprg

Description: Convert a restricted existential quantification over a pair to a disjunction. (Contributed by NM, 17-Sep-2011) (Revised by Mario Carneiro, 23-Apr-2015) Avoid ax-10 , ax-12 . (Revised by Gino Giotto, 30-Sep-2024)

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

Proof

Step	Hyp	Ref	Expression
1		ralprg.1	|- ( x = A -> ( ph <-> ps ) )
2		ralprg.2	|- ( x = B -> ( ph <-> ch ) )
3	1	notbid	|- ( x = A -> ( -. ph <-> -. ps ) )
4	2	notbid	|- ( x = B -> ( -. ph <-> -. ch ) )
5	3 4	ralprg	|- ( ( A e. V /\ B e. W ) -> ( A. x e. { A , B } -. ph <-> ( -. ps /\ -. ch ) ) )
6		ralnex	|- ( A. x e. { A , B } -. ph <-> -. E. x e. { A , B } ph )
7		pm4.56	|- ( ( -. ps /\ -. ch ) <-> -. ( ps \/ ch ) )
8	6 7	bibi12i	|- ( ( A. x e. { A , B } -. ph <-> ( -. ps /\ -. ch ) ) <-> ( -. E. x e. { A , B } ph <-> -. ( ps \/ ch ) ) )
9		notbi	|- ( ( E. x e. { A , B } ph <-> ( ps \/ ch ) ) <-> ( -. E. x e. { A , B } ph <-> -. ( ps \/ ch ) ) )
10	8 9	sylbb2	|- ( ( A. x e. { A , B } -. ph <-> ( -. ps /\ -. ch ) ) -> ( E. x e. { A , B } ph <-> ( ps \/ ch ) ) )
11	5 10	syl	|- ( ( A e. V /\ B e. W ) -> ( E. x e. { A , B } ph <-> ( ps \/ ch ) ) )