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 & C \imp D in F holds

A \imp (B \imp D) 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 & C \imp D in F holds

A \imp (B \imp D) 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 & C \imp D in F holds

A \imp (B \imp D) 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 & C \imp D in F holds

A \imp (B \imp D) in F

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

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

(A \imp (B \imp C)) \imp ((A \and B) \imp C) in F by Th48;

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

then ( (A \and B) \imp D in F & ((A \and B) \imp D) \imp (A \imp (B \imp D)) in F ) by A1, Th45, Th47;

hence A \imp (B \imp D) in F by 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 & C \imp D in F holds

A \imp (B \imp D) 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 & C \imp D in F holds

A \imp (B \imp D) 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 & C \imp D in F holds

A \imp (B \imp D) 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 & C \imp D in F holds

A \imp (B \imp D) in F

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

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

(A \imp (B \imp C)) \imp ((A \and B) \imp C) in F by Th48;

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

then ( (A \and B) \imp D in F & ((A \and B) \imp D) \imp (A \imp (B \imp D)) in F ) by A1, Th45, Th47;

hence A \imp (B \imp D) in F by Def38; :: thesis: verum