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 holds (A \imp (B \imp C)) \imp ((A \and B) \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 holds (A \imp (B \imp C)) \imp ((A \and B) \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 holds (A \imp (B \imp C)) \imp ((A \and B) \imp C) in F

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

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

A1: ((A \and B) \imp B) \imp ((B \imp C) \imp ((A \and B) \imp C)) in F by Th39;

(A \and B) \imp B in F by Def38;

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

then A2: A \imp ((B \imp C) \imp ((A \and B) \imp C)) in F by Th44;

A3: (A \imp ((A \and B) \imp C)) \imp ((A \and B) \imp (A \imp C)) in F by Th41;

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

then (A \imp (B \imp C)) \imp (A \imp ((A \and B) \imp C)) in F by A2, Def38;

then A4: (A \imp (B \imp C)) \imp ((A \and B) \imp (A \imp C)) in F by A3, Th45;

A5: (A \and B) \imp A in F by Def38;

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

then ((A \and B) \imp (A \imp C)) \imp ((A \and B) \imp C) in F by A5, Th46;

hence (A \imp (B \imp C)) \imp ((A \and B) \imp C) in F by A4, Th45; :: thesis: verum

for L being language MSAlgebra over S

for F being PC-theory of L

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

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

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

A1: ((A \and B) \imp B) \imp ((B \imp C) \imp ((A \and B) \imp C)) in F by Th39;

(A \and B) \imp B in F by Def38;

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

then A2: A \imp ((B \imp C) \imp ((A \and B) \imp C)) in F by Th44;

A3: (A \imp ((A \and B) \imp C)) \imp ((A \and B) \imp (A \imp C)) in F by Th41;

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

then (A \imp (B \imp C)) \imp (A \imp ((A \and B) \imp C)) in F by A2, Def38;

then A4: (A \imp (B \imp C)) \imp ((A \and B) \imp (A \imp C)) in F by A3, Th45;

A5: (A \and B) \imp A in F by Def38;

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

then ((A \and B) \imp (A \imp C)) \imp ((A \and B) \imp C) in F by A5, Th46;

hence (A \imp (B \imp C)) \imp ((A \and B) \imp C) in F by A4, Th45; :: thesis: verum