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 st A \iff B in F holds

B \iff 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 st A \iff B in F holds

B \iff 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 st A \iff B in F holds

B \iff A in F

let F be PC-theory of L; :: thesis: for A, B being Formula of L st A \iff B in F holds

B \iff A in F

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

assume A1: A \iff B in F ; :: thesis: B \iff A in F

( (A \iff B) \imp (A \imp B) in F & (A \iff B) \imp (B \imp A) in F ) by Def38;

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

then A2: (B \imp A) \and (A \imp B) in F by Th35;

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

hence B \iff 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 st A \iff B in F holds

B \iff 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 st A \iff B in F holds

B \iff 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 st A \iff B in F holds

B \iff A in F

let F be PC-theory of L; :: thesis: for A, B being Formula of L st A \iff B in F holds

B \iff A in F

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

assume A1: A \iff B in F ; :: thesis: B \iff A in F

( (A \iff B) \imp (A \imp B) in F & (A \iff B) \imp (B \imp A) in F ) by Def38;

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

then A2: (B \imp A) \and (A \imp B) in F by Th35;

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

hence B \iff A in F by A2, Def38; :: thesis: verum