Theorem reu6i

Description: A condition which implies existential uniqueness. (Contributed by Mario Carneiro, 2-Oct-2015)

	Ref	Expression
Assertion	reu6i	|- ( ( B e. A /\ A. x e. A ( ph <-> x = B ) ) -> E! x e. A ph )

Proof

Step	Hyp	Ref	Expression
1		eqeq2	|- ( y = B -> ( x = y <-> x = B ) )
2	1	bibi2d	|- ( y = B -> ( ( ph <-> x = y ) <-> ( ph <-> x = B ) ) )
3	2	ralbidv	|- ( y = B -> ( A. x e. A ( ph <-> x = y ) <-> A. x e. A ( ph <-> x = B ) ) )
4	3	rspcev	|- ( ( B e. A /\ A. x e. A ( ph <-> x = B ) ) -> E. y e. A A. x e. A ( ph <-> x = y ) )
5		reu6	|- ( E! x e. A ph <-> E. y e. A A. x e. A ( ph <-> x = y ) )
6	4 5	sylibr	|- ( ( B e. A /\ A. x e. A ( ph <-> x = B ) ) -> E! x e. A ph )