Theorem rexss

Description: Restricted existential quantification on a subset in terms of superset. (Contributed by Stefan O'Rear, 3-Apr-2015)

	Ref	Expression
Assertion	rexss	|- ( A C_ B -> ( E. x e. A ph <-> E. x e. B ( x e. A /\ ph ) ) )

Proof

Step	Hyp	Ref	Expression
1		ssel	|- ( A C_ B -> ( x e. A -> x e. B ) )
2	1	pm4.71rd	|- ( A C_ B -> ( x e. A <-> ( x e. B /\ x e. A ) ) )
3	2	anbi1d	|- ( A C_ B -> ( ( x e. A /\ ph ) <-> ( ( x e. B /\ x e. A ) /\ ph ) ) )
4		anass	|- ( ( ( x e. B /\ x e. A ) /\ ph ) <-> ( x e. B /\ ( x e. A /\ ph ) ) )
5	3 4	bitrdi	|- ( A C_ B -> ( ( x e. A /\ ph ) <-> ( x e. B /\ ( x e. A /\ ph ) ) ) )
6	5	rexbidv2	|- ( A C_ B -> ( E. x e. A ph <-> E. x e. B ( x e. A /\ ph ) ) )