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, D being Formula of L st A \imp (B \imp C) in F & D \imp B in F holds

A \imp (D \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, D being Formula of L st A \imp (B \imp C) in F & D \imp B in F holds

A \imp (D \imp C) in F

let L be language MSAlgebra over S; :: thesis: for F being PC-theory of L

for A, B, C, D being Formula of L st A \imp (B \imp C) in F & D \imp B in F holds

A \imp (D \imp C) in F

let F be PC-theory of L; :: thesis: for A, B, C, D being Formula of L st A \imp (B \imp C) in F & D \imp B in F holds

A \imp (D \imp C) in F

let A, B, C, D be Formula of L; :: thesis: ( A \imp (B \imp C) in F & D \imp B in F implies A \imp (D \imp C) in F )

assume A1: ( A \imp (B \imp C) in F & D \imp B in F ) ; :: thesis: A \imp (D \imp C) in F

(A \imp (B \imp C)) \imp (B \imp (A \imp C)) in F by Th41;

then B \imp (A \imp C) in F by A1, Def38;

then A2: D \imp (A \imp C) in F by A1, Th45;

(D \imp (A \imp C)) \imp (A \imp (D \imp C)) in F by Th41;

hence A \imp (D \imp C) in F by A2, Def38; :: thesis: verum

for L being language MSAlgebra over S

for F being PC-theory of L

for A, B, C, D being Formula of L st A \imp (B \imp C) in F & D \imp B in F holds

A \imp (D \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, D being Formula of L st A \imp (B \imp C) in F & D \imp B in F holds

A \imp (D \imp C) in F

let L be language MSAlgebra over S; :: thesis: for F being PC-theory of L

for A, B, C, D being Formula of L st A \imp (B \imp C) in F & D \imp B in F holds

A \imp (D \imp C) in F

let F be PC-theory of L; :: thesis: for A, B, C, D being Formula of L st A \imp (B \imp C) in F & D \imp B in F holds

A \imp (D \imp C) in F

let A, B, C, D be Formula of L; :: thesis: ( A \imp (B \imp C) in F & D \imp B in F implies A \imp (D \imp C) in F )

assume A1: ( A \imp (B \imp C) in F & D \imp B in F ) ; :: thesis: A \imp (D \imp C) in F

(A \imp (B \imp C)) \imp (B \imp (A \imp C)) in F by Th41;

then B \imp (A \imp C) in F by A1, Def38;

then A2: D \imp (A \imp C) in F by A1, Th45;

(D \imp (A \imp C)) \imp (A \imp (D \imp C)) in F by Th41;

hence A \imp (D \imp C) in F by A2, Def38; :: thesis: verum