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Metamath Proof Explorer

Theorem isgbe

Description: The predicate "is an even Goldbach number". An even Goldbach number is an even integer having a Goldbach partition, i.e. which can be written as a sum of two odd primes. (Contributed by AV, 20-Jul-2020)

Ref Expression
Assertion isgbe 
|- ( Z e. GoldbachEven <-> ( Z e. Even /\ E. p e. Prime E. q e. Prime ( p e. Odd /\ q e. Odd /\ Z = ( p + q ) ) ) )

Proof

Step Hyp Ref Expression
1 eqeq1 
 |-  ( z = Z -> ( z = ( p + q ) <-> Z = ( p + q ) ) )
2 1 3anbi3d 
 |-  ( z = Z -> ( ( p e. Odd /\ q e. Odd /\ z = ( p + q ) ) <-> ( p e. Odd /\ q e. Odd /\ Z = ( p + q ) ) ) )
3 2 2rexbidv 
 |-  ( z = Z -> ( E. p e. Prime E. q e. Prime ( p e. Odd /\ q e. Odd /\ z = ( p + q ) ) <-> E. p e. Prime E. q e. Prime ( p e. Odd /\ q e. Odd /\ Z = ( p + q ) ) ) )
4 df-gbe 
 |-  GoldbachEven = { z e. Even | E. p e. Prime E. q e. Prime ( p e. Odd /\ q e. Odd /\ z = ( p + q ) ) }
5 3 4 elrab2 
 |-  ( Z e. GoldbachEven <-> ( Z e. Even /\ E. p e. Prime E. q e. Prime ( p e. Odd /\ q e. Odd /\ Z = ( p + q ) ) ) )