Theorem rexdifsn

Description: Restricted existential quantification over a set with an element removed. (Contributed by NM, 4-Feb-2015)

	Ref	Expression
Assertion	rexdifsn	|- ( E. x e. ( A \ { B } ) ph <-> E. x e. A ( x =/= B /\ ph ) )

Proof

Step	Hyp	Ref	Expression
1		eldifsn	|- ( x e. ( A \ { B } ) <-> ( x e. A /\ x =/= B ) )
2	1	anbi1i	|- ( ( x e. ( A \ { B } ) /\ ph ) <-> ( ( x e. A /\ x =/= B ) /\ ph ) )
3		anass	|- ( ( ( x e. A /\ x =/= B ) /\ ph ) <-> ( x e. A /\ ( x =/= B /\ ph ) ) )
4	2 3	bitri	|- ( ( x e. ( A \ { B } ) /\ ph ) <-> ( x e. A /\ ( x =/= B /\ ph ) ) )
5	4	rexbii2	|- ( E. x e. ( A \ { B } ) ph <-> E. x e. A ( x =/= B /\ ph ) )