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

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

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

set qp = B \imp (A \and B);

set pr = ((A \and B) \imp C) \imp (B \imp C);

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

A2: A \imp (B \imp (A \and B)) in F by Def38;

A \imp ((B \imp (A \and B)) \imp (((A \and B) \imp C) \imp (B \imp C))) in F by Th44, Th39;

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

then A3: A \imp (((A \and B) \imp C) \imp (B \imp C)) in F by A2, Def38;

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

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

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

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

set qp = B \imp (A \and B);

set pr = ((A \and B) \imp C) \imp (B \imp C);

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

A2: A \imp (B \imp (A \and B)) in F by Def38;

A \imp ((B \imp (A \and B)) \imp (((A \and B) \imp C) \imp (B \imp C))) in F by Th44, Th39;

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

then A3: A \imp (((A \and B) \imp C) \imp (B \imp C)) in F by A2, Def38;

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

hence ((A \and B) \imp C) \imp (A \imp (B \imp C)) in F by A3, Def38; :: thesis: verum