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 holds ((A \and B) \and C) \iff (A \and (B \and 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 holds ((A \and B) \and C) \iff (A \and (B \and 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 holds ((A \and B) \and C) \iff (A \and (B \and C)) in F
let F be PC-theory of L; for A, B, C being Formula of L holds ((A \and B) \and C) \iff (A \and (B \and C)) in F
let A, B, C be Formula of L; ((A \and B) \and C) \iff (A \and (B \and C)) in F
A1:
(((A \and B) \and C) \imp A) \imp ((((A \and B) \and C) \imp (B \and C)) \imp (((A \and B) \and C) \imp (A \and (B \and C)))) in F
by Th49;
( (A \and B) \imp A in F & ((A \and B) \and C) \imp (A \and B) in F )
by Def38;
then
((A \and B) \and C) \imp A in F
by Th45;
then A2:
(((A \and B) \and C) \imp (B \and C)) \imp (((A \and B) \and C) \imp (A \and (B \and C))) in F
by A1, Def38;
( (A \and B) \imp B in F & C \imp C in F )
by Def38, Th34;
then
((A \and B) \and C) \imp (B \and C) in F
by Th72;
then A3:
((A \and B) \and C) \imp (A \and (B \and C)) in F
by A2, Def38;
A4:
((A \and (B \and C)) \imp (A \and B)) \imp (((A \and (B \and C)) \imp C) \imp ((A \and (B \and C)) \imp ((A \and B) \and C))) in F
by Th49;
( (B \and C) \imp B in F & A \imp A in F )
by Def38, Th34;
then
(A \and (B \and C)) \imp (A \and B) in F
by Th72;
then A5:
((A \and (B \and C)) \imp C) \imp ((A \and (B \and C)) \imp ((A \and B) \and C)) in F
by A4, Def38;
( (B \and C) \imp C in F & (A \and (B \and C)) \imp (B \and C) in F )
by Def38;
then
(A \and (B \and C)) \imp C in F
by Th45;
then
(A \and (B \and C)) \imp ((A \and B) \and C) in F
by A5, Def38;
hence
((A \and B) \and C) \iff (A \and (B \and C)) in F
by A3, Th43; verum