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) \imp (\not (A \and (\not B))) 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) \imp (\not (A \and (\not B))) 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) \imp (\not (A \and (\not B))) in F
let F be PC-theory of L; :: thesis: for A, B being Formula of L holds (A \imp B) \imp (\not (A \and (\not B))) in F
let A, B be Formula of L; :: thesis: (A \imp B) \imp (\not (A \and (\not B))) in F
A1: (A \imp B) \imp ((\not A) \or B) in F by Th82;
( (\not A) \imp (\not A) in F & B \imp (\not (\not B)) in F ) by Th64, Th34;
then ((\not A) \or B) \imp ((\not A) \or (\not (\not B))) in F by Th59;
then A2: (A \imp B) \imp ((\not A) \or (\not (\not B))) in F by A1, Th45;
((\not A) \or (\not (\not B))) \imp (\not (A \and (\not B))) in F by Th73;
hence (A \imp B) \imp (\not (A \and (\not B))) in F by A2, Th45; :: thesis: verum