Metamath Proof Explorer

Theorem elisset

Description: An element of a class exists. (Contributed by NM, 1-May-1995) Reduce dependencies on axioms. (Revised by BJ, 29-Apr-2019)

Ref Expression
Assertion elisset 
|- ( A e. V -> E. x x = A )

Proof

Step Hyp Ref Expression
1 exsimpl 
 |-  ( E. y ( y = A /\ y e. V ) -> E. y y = A )
2 dfclel 
 |-  ( A e. V <-> E. y ( y = A /\ y e. V ) )
3 eqeq1 
 |-  ( x = y -> ( x = A <-> y = A ) )
4 3 cbvexvw 
 |-  ( E. x x = A <-> E. y y = A )
5 1 2 4 3imtr4i 
 |-  ( A e. V -> E. x x = A )