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Metamath Proof Explorer

Theorem isgbow

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

Ref Expression
Assertion isgbow 
|- ( Z e. GoldbachOddW <-> ( Z e. Odd /\ E. p e. Prime E. q e. Prime E. r e. Prime Z = ( ( p + q ) + r ) ) )

Proof

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