Metamath Proof Explorer


Theorem eusv2

Description: Two ways to express single-valuedness of a class expression A ( x ) . (Contributed by NM, 15-Oct-2010) (Proof shortened by Mario Carneiro, 18-Nov-2016)

Ref Expression
Hypothesis eusv2.1
|- A e. _V
Assertion eusv2
|- ( E! y E. x y = A <-> E! y A. x y = A )

Proof

Step Hyp Ref Expression
1 eusv2.1
 |-  A e. _V
2 1 eusv2nf
 |-  ( E! y E. x y = A <-> F/_ x A )
3 eusvnfb
 |-  ( E! y A. x y = A <-> ( F/_ x A /\ A e. _V ) )
4 1 3 mpbiran2
 |-  ( E! y A. x y = A <-> F/_ x A )
5 2 4 bitr4i
 |-  ( E! y E. x y = A <-> E! y A. x y = A )