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 ) )