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 st A \imp B in F holds

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

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

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

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

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

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

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

then (A \imp B) \imp ((C \imp A) \imp (C \imp B)) in F by Th38;

hence ( A \imp B in F implies (C \imp A) \imp (C \imp B) in F ) by 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 st A \imp B in F holds

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

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

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

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

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

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

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

then (A \imp B) \imp ((C \imp A) \imp (C \imp B)) in F by Th38;

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