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 being Formula of L holds (A \and B) \imp (B \and A) 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 being Formula of L holds (A \and B) \imp (B \and A) in F

let L be language MSAlgebra over S; :: thesis: for F being PC-theory of L

for A, B being Formula of L holds (A \and B) \imp (B \and A) in F

let F be PC-theory of L; :: thesis: for A, B being Formula of L holds (A \and B) \imp (B \and A) in F

let A, B be Formula of L; :: thesis: (A \and B) \imp (B \and A) in F

set P = A \and B;

A1: (A \and B) \imp B in F by Def38;

A2: (A \and B) \imp A in F by Def38;

((A \and B) \imp B) \imp (((A \and B) \imp A) \imp ((A \and B) \imp (B \and A))) in F by Th49;

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

hence (A \and B) \imp (B \and A) in F by A2, Def38; :: thesis: verum

for L being language MSAlgebra over S

for F being PC-theory of L

for A, B being Formula of L holds (A \and B) \imp (B \and A) 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 being Formula of L holds (A \and B) \imp (B \and A) in F

let L be language MSAlgebra over S; :: thesis: for F being PC-theory of L

for A, B being Formula of L holds (A \and B) \imp (B \and A) in F

let F be PC-theory of L; :: thesis: for A, B being Formula of L holds (A \and B) \imp (B \and A) in F

let A, B be Formula of L; :: thesis: (A \and B) \imp (B \and A) in F

set P = A \and B;

A1: (A \and B) \imp B in F by Def38;

A2: (A \and B) \imp A in F by Def38;

((A \and B) \imp B) \imp (((A \and B) \imp A) \imp ((A \and B) \imp (B \and A))) in F by Th49;

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

hence (A \and B) \imp (B \and A) in F by A2, Def38; :: thesis: verum