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 \iff C in F holds
A \iff 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 \iff C in F holds
A \iff 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 \iff C in F holds
A \iff 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 \iff C in F holds
A \iff C in F
let A, B, C be Formula of L; :: thesis: ( A \iff B in F & B \iff C in F implies A \iff C in F )
assume A1: ( A \iff B in F & B \iff C in F ) ; :: thesis: A \iff 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 & B \imp A in F & C \imp B in F & B \imp C in F ) by A1, Def38;
then ( A \imp C in F & C \imp A in F ) by Th45;
then A2: (A \imp C) \and (C \imp A) in F by Th35;
((A \imp C) \and (C \imp A)) \imp (A \iff C) in F by Def38;
hence A \iff C in F by A2, Def38; :: thesis: verum