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 \iff B in F iff (\not (\not A)) \iff B 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 \iff B in F iff (\not (\not A)) \iff B 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 \iff B in F iff (\not (\not A)) \iff B in F )

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

( A \iff B in F iff (\not (\not A)) \iff B in F )

let A, B be Formula of L; :: thesis: ( A \iff B in F iff (\not (\not A)) \iff B in F )

( (\not (\not A)) \imp A in F & A \imp (\not (\not A)) in F ) by Th64, Th65;

then ( ((\not (\not A)) \imp A) \and (A \imp (\not (\not A))) in F & (((\not (\not A)) \imp A) \and (A \imp (\not (\not A)))) \imp ((\not (\not A)) \iff A) in F ) by Th35, Def38;

then A1: (\not (\not A)) \iff A in F by Def38;

then A \iff (\not (\not A)) in F by Th90;

hence ( A \iff B in F iff (\not (\not A)) \iff B in F ) by A1, Th91; :: 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 \iff B in F iff (\not (\not A)) \iff B 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 \iff B in F iff (\not (\not A)) \iff B 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 \iff B in F iff (\not (\not A)) \iff B in F )

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

( A \iff B in F iff (\not (\not A)) \iff B in F )

let A, B be Formula of L; :: thesis: ( A \iff B in F iff (\not (\not A)) \iff B in F )

( (\not (\not A)) \imp A in F & A \imp (\not (\not A)) in F ) by Th64, Th65;

then ( ((\not (\not A)) \imp A) \and (A \imp (\not (\not A))) in F & (((\not (\not A)) \imp A) \and (A \imp (\not (\not A)))) \imp ((\not (\not A)) \iff A) in F ) by Th35, Def38;

then A1: (\not (\not A)) \iff A in F by Def38;

then A \iff (\not (\not A)) in F by Th90;

hence ( A \iff B in F iff (\not (\not A)) \iff B in F ) by A1, Th91; :: thesis: verum