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Metamath Proof Explorer

Theorem elrabf

Description: Membership in a restricted class abstraction, using implicit substitution. This version has bound-variable hypotheses in place of distinct variable restrictions. (Contributed by NM, 21-Sep-2003)

Ref Expression
Hypotheses elrabf.1 
|- F/_ x A
elrabf.2 
|- F/_ x B
elrabf.3 
|- F/ x ps
elrabf.4 
|- ( x = A -> ( ph <-> ps ) )
Assertion elrabf 
|- ( A e. { x e. B | ph } <-> ( A e. B /\ ps ) )

Proof

Step Hyp Ref Expression
1 elrabf.1 
 |-  F/_ x A
2 elrabf.2 
 |-  F/_ x B
3 elrabf.3 
 |-  F/ x ps
4 elrabf.4 
 |-  ( x = A -> ( ph <-> ps ) )
5 elex 
 |-  ( A e. { x e. B | ph } -> A e. _V )
6 elex 
 |-  ( A e. B -> A e. _V )
7 6 adantr 
 |-  ( ( A e. B /\ ps ) -> A e. _V )
8 df-rab 
 |-  { x e. B | ph } = { x | ( x e. B /\ ph ) }
9 8 eleq2i 
 |-  ( A e. { x e. B | ph } <-> A e. { x | ( x e. B /\ ph ) } )
10 1 2 nfel 
 |-  F/ x A e. B
11 10 3 nfan 
 |-  F/ x ( A e. B /\ ps )
12 eleq1 
 |-  ( x = A -> ( x e. B <-> A e. B ) )
13 12 4 anbi12d 
 |-  ( x = A -> ( ( x e. B /\ ph ) <-> ( A e. B /\ ps ) ) )
14 1 11 13 elabgf 
 |-  ( A e. _V -> ( A e. { x | ( x e. B /\ ph ) } <-> ( A e. B /\ ps ) ) )
15 9 14 bitrid 
 |-  ( A e. _V -> ( A e. { x e. B | ph } <-> ( A e. B /\ ps ) ) )
16 5 7 15 pm5.21nii 
 |-  ( A e. { x e. B | ph } <-> ( A e. B /\ ps ) )