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Metamath Proof Explorer

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 ) ) )