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

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