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Metamath Proof Explorer

Theorem isgbo

Description: The predicate "is an odd Goldbach number". An odd Goldbach number is an odd integer having a Goldbach partition, i.e. which can be written as sum of three odd primes. (Contributed by AV, 26-Jul-2020)

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

Proof

Step Hyp Ref Expression
1 eqeq1 
 |-  ( z = Z -> ( z = ( ( p + q ) + r ) <-> Z = ( ( p + q ) + r ) ) )
2 1 anbi2d 
 |-  ( z = Z -> ( ( ( p e. Odd /\ q e. Odd /\ r e. Odd ) /\ z = ( ( p + q ) + r ) ) <-> ( ( p e. Odd /\ q e. Odd /\ r e. Odd ) /\ Z = ( ( p + q ) + r ) ) ) )
3 2 rexbidv 
 |-  ( z = Z -> ( E. r e. Prime ( ( p e. Odd /\ q e. Odd /\ r e. Odd ) /\ z = ( ( p + q ) + r ) ) <-> E. r e. Prime ( ( p e. Odd /\ q e. Odd /\ r e. Odd ) /\ Z = ( ( p + q ) + r ) ) ) )
4 3 2rexbidv 
 |-  ( z = Z -> ( E. p e. Prime E. q e. Prime E. r e. Prime ( ( p e. Odd /\ q e. Odd /\ r e. Odd ) /\ z = ( ( p + q ) + r ) ) <-> E. p e. Prime E. q e. Prime E. r e. Prime ( ( p e. Odd /\ q e. Odd /\ r e. Odd ) /\ Z = ( ( p + q ) + r ) ) ) )
5 df-gbo 
 |-  GoldbachOdd = { z e. Odd | E. p e. Prime E. q e. Prime E. r e. Prime ( ( p e. Odd /\ q e. Odd /\ r e. Odd ) /\ z = ( ( p + q ) + r ) ) }
6 4 5 elrab2 
 |-  ( Z e. GoldbachOdd <-> ( Z e. Odd /\ E. p e. Prime E. q e. Prime E. r e. Prime ( ( p e. Odd /\ q e. Odd /\ r e. Odd ) /\ Z = ( ( p + q ) + r ) ) ) )