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Metamath Proof Explorer

Theorem ssrab

Description: Subclass of a restricted class abstraction. (Contributed by NM, 16-Aug-2006)

Ref Expression
Assertion ssrab 
|- ( B C_ { x e. A | ph } <-> ( B C_ A /\ A. x e. B ph ) )

Proof

Step Hyp Ref Expression
1 df-rab 
 |-  { x e. A | ph } = { x | ( x e. A /\ ph ) }
2 1 sseq2i 
 |-  ( B C_ { x e. A | ph } <-> B C_ { x | ( x e. A /\ ph ) } )
3 ssab 
 |-  ( B C_ { x | ( x e. A /\ ph ) } <-> A. x ( x e. B -> ( x e. A /\ ph ) ) )
4 dfss3 
 |-  ( B C_ A <-> A. x e. B x e. A )
5 4 anbi1i 
 |-  ( ( B C_ A /\ A. x e. B ph ) <-> ( A. x e. B x e. A /\ A. x e. B ph ) )
6 r19.26 
 |-  ( A. x e. B ( x e. A /\ ph ) <-> ( A. x e. B x e. A /\ A. x e. B ph ) )
7 df-ral 
 |-  ( A. x e. B ( x e. A /\ ph ) <-> A. x ( x e. B -> ( x e. A /\ ph ) ) )
8 5 6 7 3bitr2ri 
 |-  ( A. x ( x e. B -> ( x e. A /\ ph ) ) <-> ( B C_ A /\ A. x e. B ph ) )
9 2 3 8 3bitri 
 |-  ( B C_ { x e. A | ph } <-> ( B C_ A /\ A. x e. B ph ) )