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 ) ) ) )