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Metamath Proof Explorer

Theorem elunirab

Description: Membership in union of a class abstraction. (Contributed by NM, 4-Oct-2006)

Ref Expression
Assertion elunirab 
|- ( A e. U. { x e. B | ph } <-> E. x e. B ( A e. x /\ ph ) )

Proof

Step Hyp Ref Expression
1 eluniab 
 |-  ( A e. U. { x | ( x e. B /\ ph ) } <-> E. x ( A e. x /\ ( x e. B /\ ph ) ) )
2 df-rab 
 |-  { x e. B | ph } = { x | ( x e. B /\ ph ) }
3 2 unieqi 
 |-  U. { x e. B | ph } = U. { x | ( x e. B /\ ph ) }
4 3 eleq2i 
 |-  ( A e. U. { x e. B | ph } <-> A e. U. { x | ( x e. B /\ ph ) } )
5 df-rex 
 |-  ( E. x e. B ( A e. x /\ ph ) <-> E. x ( x e. B /\ ( A e. x /\ ph ) ) )
6 an12 
 |-  ( ( x e. B /\ ( A e. x /\ ph ) ) <-> ( A e. x /\ ( x e. B /\ ph ) ) )
7 6 exbii 
 |-  ( E. x ( x e. B /\ ( A e. x /\ ph ) ) <-> E. x ( A e. x /\ ( x e. B /\ ph ) ) )
8 5 7 bitri 
 |-  ( E. x e. B ( A e. x /\ ph ) <-> E. x ( A e. x /\ ( x e. B /\ ph ) ) )
9 1 4 8 3bitr4i 
 |-  ( A e. U. { x e. B | ph } <-> E. x e. B ( A e. x /\ ph ) )