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, C being Formula of L st A \imp (B \imp C) in F holds
B \imp (A \imp C) 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, C being Formula of L st A \imp (B \imp C) in F holds
B \imp (A \imp C) in F
let L be language MSAlgebra over S; :: thesis: for F being PC-theory of L
for A, B, C being Formula of L st A \imp (B \imp C) in F holds
B \imp (A \imp C) in F
let F be PC-theory of L; :: thesis: for A, B, C being Formula of L st A \imp (B \imp C) in F holds
B \imp (A \imp C) in F
let A, B, C be Formula of L; :: thesis: ( A \imp (B \imp C) in F implies B \imp (A \imp C) in F )
assume A1: A \imp (B \imp C) in F ; :: thesis: B \imp (A \imp C) in F
A2: ((A \imp B) \imp (A \imp C)) \imp ((B \imp (A \imp B)) \imp (B \imp (A \imp C))) in F by Th37;
(A \imp (B \imp C)) \imp ((A \imp B) \imp (A \imp C)) in F by Def38;
then (A \imp B) \imp (A \imp C) in F by A1, Def38;
then A3: (B \imp (A \imp B)) \imp (B \imp (A \imp C)) in F by A2, Def38;
B \imp (A \imp B) in F by Def38;
hence B \imp (A \imp C) in F by A3, Def38; :: thesis: verum