Theorem sbgoldbmb

Description: The strong binary Goldbach conjecture and the modern version of the original formulation of the Goldbach conjecture are equivalent. (Contributed by AV, 26-Dec-2021)

	Ref	Expression
Assertion	sbgoldbmb	|- ( A. n e. Even ( 4 < n -> n e. GoldbachEven ) <-> A. n e. ( ZZ>= ` 6 ) E. p e. Prime E. q e. Prime E. r e. Prime n = ( ( p + q ) + r ) )

Proof

Step	Hyp	Ref	Expression
1		sbgoldbm	|- ( A. n e. Even ( 4 < n -> n e. GoldbachEven ) -> A. n e. ( ZZ>= ` 6 ) E. p e. Prime E. q e. Prime E. r e. Prime n = ( ( p + q ) + r ) )
2		mogoldbb	|- ( A. n e. ( ZZ>= ` 6 ) E. p e. Prime E. q e. Prime E. r e. Prime n = ( ( p + q ) + r ) -> A. n e. Even ( 2 < n -> E. p e. Prime E. q e. Prime n = ( p + q ) ) )
3		sbgoldbalt	|- ( A. n e. Even ( 4 < n -> n e. GoldbachEven ) <-> A. n e. Even ( 2 < n -> E. p e. Prime E. q e. Prime n = ( p + q ) ) )
4	2 3	sylibr	|- ( A. n e. ( ZZ>= ` 6 ) E. p e. Prime E. q e. Prime E. r e. Prime n = ( ( p + q ) + r ) -> A. n e. Even ( 4 < n -> n e. GoldbachEven ) )
5	1 4	impbii	|- ( A. n e. Even ( 4 < n -> n e. GoldbachEven ) <-> A. n e. ( ZZ>= ` 6 ) E. p e. Prime E. q e. Prime E. r e. Prime n = ( ( p + q ) + r ) )