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 \iff B in F & B \imp C in F holds

A \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 st A \iff B in F & B \imp C in F holds

A \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 st A \iff B in F & B \imp C in F holds

A \imp C in F

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

A \imp C in F

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

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

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

then A \imp B in F by A1, Def38;

hence A \imp C 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 st A \iff B in F & B \imp C in F holds

A \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 st A \iff B in F & B \imp C in F holds

A \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 st A \iff B in F & B \imp C in F holds

A \imp C in F

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

A \imp C in F

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

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

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

then A \imp B in F by A1, Def38;

hence A \imp C in F by A1, Th45; :: thesis: verum