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

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

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

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

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

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

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

A1: ( (\not A) \imp B in F iff (\not B) \imp (\not (\not A)) in F ) by Th58;

A2: ( (\not B) \imp A in F iff (\not A) \imp (\not (\not B)) in F ) by Th58;

( (\not (\not A)) \imp A in F & (\not (\not B)) \imp B in F ) by Th65;

hence ( (\not A) \imp B in F iff (\not B) \imp A in F ) by A1, A2, Th45; :: thesis: verum

for L being language MSAlgebra over S

for F being PC-theory of L

for A, B being Formula of L holds

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

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

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

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

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

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

A1: ( (\not A) \imp B in F iff (\not B) \imp (\not (\not A)) in F ) by Th58;

A2: ( (\not B) \imp A in F iff (\not A) \imp (\not (\not B)) in F ) by Th58;

( (\not (\not A)) \imp A in F & (\not (\not B)) \imp B in F ) by Th65;

hence ( (\not A) \imp B in F iff (\not B) \imp A in F ) by A1, A2, Th45; :: thesis: verum