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, D being Formula of L st A \imp (B \imp C) in F & C \imp D in F holds
A \imp (B \imp D) 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, D being Formula of L st A \imp (B \imp C) in F & C \imp D in F holds
A \imp (B \imp D) in F
let L be language MSAlgebra over S; for F being PC-theory of L
for A, B, C, D being Formula of L st A \imp (B \imp C) in F & C \imp D in F holds
A \imp (B \imp D) in F
let F be PC-theory of L; for A, B, C, D being Formula of L st A \imp (B \imp C) in F & C \imp D in F holds
A \imp (B \imp D) in F
let A, B, C, D be Formula of L; ( A \imp (B \imp C) in F & C \imp D in F implies A \imp (B \imp D) in F )
assume A1:
( A \imp (B \imp C) in F & C \imp D in F )
; A \imp (B \imp D) in F
(A \imp (B \imp C)) \imp ((A \and B) \imp C) in F
by Th48;
then
(A \and B) \imp C in F
by A1, Def38;
then
( (A \and B) \imp D in F & ((A \and B) \imp D) \imp (A \imp (B \imp D)) in F )
by A1, Th45, Th47;
hence
A \imp (B \imp D) in F
by Def38; verum