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Metamath Proof Explorer

Theorem eluniab

Description: Membership in union of a class abstraction. (Contributed by NM, 11-Aug-1994) (Revised by Mario Carneiro, 14-Nov-2016)

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

Proof

Step Hyp Ref Expression
1 eluni 
 |-  ( A e. U. { x | ph } <-> E. y ( A e. y /\ y e. { x | ph } ) )
2 nfv 
 |-  F/ x A e. y
3 nfsab1 
 |-  F/ x y e. { x | ph }
4 2 3 nfan 
 |-  F/ x ( A e. y /\ y e. { x | ph } )
5 nfv 
 |-  F/ y ( A e. x /\ ph )
6 eleq2w 
 |-  ( y = x -> ( A e. y <-> A e. x ) )
7 eleq1w 
 |-  ( y = x -> ( y e. { x | ph } <-> x e. { x | ph } ) )
8 abid 
 |-  ( x e. { x | ph } <-> ph )
9 7 8 bitrdi 
 |-  ( y = x -> ( y e. { x | ph } <-> ph ) )
10 6 9 anbi12d 
 |-  ( y = x -> ( ( A e. y /\ y e. { x | ph } ) <-> ( A e. x /\ ph ) ) )
11 4 5 10 cbvexv1 
 |-  ( E. y ( A e. y /\ y e. { x | ph } ) <-> E. x ( A e. x /\ ph ) )
12 1 11 bitri 
 |-  ( A e. U. { x | ph } <-> E. x ( A e. x /\ ph ) )