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) (Proof shortened by AV, 8-Apr-2023)

	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		nfv	|- F/ x ps
4		nfv	|- F/ x ch
5	3 4 1 2	rexprgf	|- ( ( A e. V /\ B e. W ) -> ( E. x e. { A , B } ph <-> ( ps \/ ch ) ) )