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

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

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

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

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

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

( B \imp (\not (\not B)) in F & A \imp (\not (\not A)) in F ) by Th64;

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

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

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

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

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

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

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

( B \imp (\not (\not B)) in F & A \imp (\not (\not A)) in F ) by Th64;

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