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Metamath Proof Explorer

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 ) )