let n be non empty Nat; 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 ; 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; 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; 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; ( 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 )
; 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; verum