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 ) ) )