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Metamath Proof Explorer

Theorem elintrab

Description: Membership in the intersection of a class abstraction. (Contributed by NM, 17-Oct-1999)

Ref Expression
Hypothesis elintab.ex 
|- A e. _V
Assertion elintrab 
|- ( A e. |^| { x e. B | ph } <-> A. x e. B ( ph -> A e. x ) )

Proof

Step Hyp Ref Expression
1 elintab.ex 
 |-  A e. _V
2 1 elintab 
 |-  ( A e. |^| { x | ( x e. B /\ ph ) } <-> A. x ( ( x e. B /\ ph ) -> A e. x ) )
3 impexp 
 |-  ( ( ( x e. B /\ ph ) -> A e. x ) <-> ( x e. B -> ( ph -> A e. x ) ) )
4 3 albii 
 |-  ( A. x ( ( x e. B /\ ph ) -> A e. x ) <-> A. x ( x e. B -> ( ph -> A e. x ) ) )
5 2 4 bitri 
 |-  ( A e. |^| { x | ( x e. B /\ ph ) } <-> A. x ( x e. B -> ( ph -> A e. x ) ) )
6 df-rab 
 |-  { x e. B | ph } = { x | ( x e. B /\ ph ) }
7 6 inteqi 
 |-  |^| { x e. B | ph } = |^| { x | ( x e. B /\ ph ) }
8 7 eleq2i 
 |-  ( A e. |^| { x e. B | ph } <-> A e. |^| { x | ( x e. B /\ ph ) } )
9 df-ral 
 |-  ( A. x e. B ( ph -> A e. x ) <-> A. x ( x e. B -> ( ph -> A e. x ) ) )
10 5 8 9 3bitr4i 
 |-  ( A e. |^| { x e. B | ph } <-> A. x e. B ( ph -> A e. x ) )