let n be non empty Nat; :: thesis: for S being non empty non void n PC-correct PCLangSignature

for L being language MSAlgebra over S

for F being PC-theory of L

for A, B being Formula of L holds

( A \imp B in F iff (\not B) \imp (\not A) in F )

let S be non empty non void n PC-correct PCLangSignature ; :: thesis: for L being language MSAlgebra over S

for F being PC-theory of L

for A, B being Formula of L holds

( A \imp B in F iff (\not B) \imp (\not A) in F )

let L be language MSAlgebra over S; :: thesis: for F being PC-theory of L

for A, B being Formula of L holds

( A \imp B in F iff (\not B) \imp (\not A) in F )

let F be PC-theory of L; :: thesis: for A, B being Formula of L holds

( A \imp B in F iff (\not B) \imp (\not A) in F )

let A, B be Formula of L; :: thesis: ( A \imp B in F iff (\not B) \imp (\not A) in F )

( ((\not B) \imp (\not A)) \imp (A \imp B) in F & (A \imp B) \imp ((\not B) \imp (\not A)) in F ) by Def38, Th57;

hence ( A \imp B in F iff (\not B) \imp (\not A) in F ) by Def38; :: thesis: verum

for L being language MSAlgebra over S

for F being PC-theory of L

for A, B being Formula of L holds

( A \imp B in F iff (\not B) \imp (\not A) in F )

let S be non empty non void n PC-correct PCLangSignature ; :: thesis: for L being language MSAlgebra over S

for F being PC-theory of L

for A, B being Formula of L holds

( A \imp B in F iff (\not B) \imp (\not A) in F )

let L be language MSAlgebra over S; :: thesis: for F being PC-theory of L

for A, B being Formula of L holds

( A \imp B in F iff (\not B) \imp (\not A) in F )

let F be PC-theory of L; :: thesis: for A, B being Formula of L holds

( A \imp B in F iff (\not B) \imp (\not A) in F )

let A, B be Formula of L; :: thesis: ( A \imp B in F iff (\not B) \imp (\not A) in F )

( ((\not B) \imp (\not A)) \imp (A \imp B) in F & (A \imp B) \imp ((\not B) \imp (\not A)) in F ) by Def38, Th57;

hence ( A \imp B in F iff (\not B) \imp (\not A) in F ) by Def38; :: thesis: verum