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

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

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

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

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

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

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

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

hence (C \imp A) \imp ((C \imp B) \imp (C \imp (A \and B))) in F by A1, 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 (C \imp A) \imp ((C \imp B) \imp (C \imp (A \and B))) 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 (C \imp A) \imp ((C \imp B) \imp (C \imp (A \and B))) 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 (C \imp A) \imp ((C \imp B) \imp (C \imp (A \and B))) in F

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

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

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

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

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

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

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

hence (C \imp A) \imp ((C \imp B) \imp (C \imp (A \and B))) in F by A1, Th45; :: thesis: verum