Theorem rexss

Description: Restricted existential quantification on a subset in terms of superset. (Contributed by Stefan O'Rear, 3-Apr-2015) Avoid axioms. (Revised by SN, 14-Oct-2025)

	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		df-ss	|- ( A C_ B <-> A. x ( x e. A -> x e. B ) )
2		pm3.41	|- ( ( x e. A -> x e. B ) -> ( ( x e. A /\ ph ) -> x e. B ) )
3	2	pm4.71rd	|- ( ( x e. A -> x e. B ) -> ( ( x e. A /\ ph ) <-> ( x e. B /\ ( x e. A /\ ph ) ) ) )
4	3	alexbii	|- ( A. x ( x e. A -> x e. B ) -> ( E. x ( x e. A /\ ph ) <-> E. x ( x e. B /\ ( x e. A /\ ph ) ) ) )
5	1 4	sylbi	|- ( A C_ B -> ( E. x ( x e. A /\ ph ) <-> E. x ( x e. B /\ ( x e. A /\ ph ) ) ) )
6		df-rex	|- ( E. x e. A ph <-> E. x ( x e. A /\ ph ) )
7		df-rex	|- ( E. x e. B ( x e. A /\ ph ) <-> E. x ( x e. B /\ ( x e. A /\ ph ) ) )
8	5 6 7	3bitr4g	|- ( A C_ B -> ( E. x e. A ph <-> E. x e. B ( x e. A /\ ph ) ) )