Metamath Proof Explorer


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 ) ) )