This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem ssintrab

Description: Subclass of the intersection of a restricted class abstraction. (Contributed by NM, 30-Jan-2015)

Ref Expression
Assertion ssintrab 
|- ( A C_ |^| { x e. B | ph } <-> A. x e. B ( ph -> A C_ x ) )

Proof

Step Hyp Ref Expression
1 df-rab 
 |-  { x e. B | ph } = { x | ( x e. B /\ ph ) }
2 1 inteqi 
 |-  |^| { x e. B | ph } = |^| { x | ( x e. B /\ ph ) }
3 2 sseq2i 
 |-  ( A C_ |^| { x e. B | ph } <-> A C_ |^| { x | ( x e. B /\ ph ) } )
4 impexp 
 |-  ( ( ( x e. B /\ ph ) -> A C_ x ) <-> ( x e. B -> ( ph -> A C_ x ) ) )
5 4 albii 
 |-  ( A. x ( ( x e. B /\ ph ) -> A C_ x ) <-> A. x ( x e. B -> ( ph -> A C_ x ) ) )
6 ssintab 
 |-  ( A C_ |^| { x | ( x e. B /\ ph ) } <-> A. x ( ( x e. B /\ ph ) -> A C_ x ) )
7 df-ral 
 |-  ( A. x e. B ( ph -> A C_ x ) <-> A. x ( x e. B -> ( ph -> A C_ x ) ) )
8 5 6 7 3bitr4i 
 |-  ( A C_ |^| { x | ( x e. B /\ ph ) } <-> A. x e. B ( ph -> A C_ x ) )
9 3 8 bitri 
 |-  ( A C_ |^| { x e. B | ph } <-> A. x e. B ( ph -> A C_ x ) )