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

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

( ((\not A) \iff (\not B)) \imp ((\not A) \imp (\not B)) in F & ((\not A) \iff (\not B)) \imp ((\not B) \imp (\not A)) in F ) by Def38;

then ( (\not A) \imp (\not B) in F & (\not B) \imp (\not A) in F & ((\not A) \imp (\not B)) \imp (B \imp A) in F & ((\not B) \imp (\not A)) \imp (A \imp B) in F ) by A2, Def38;

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

then ( (A \imp B) \and (B \imp A) in F & ((A \imp B) \and (B \imp A)) \imp (A \iff B) in F ) by Def38, Th35;

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

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

hereby :: thesis: ( (\not A) \iff (\not B) in F implies A \iff B in F )

assume A2:
(\not A) \iff (\not B) in F
; :: thesis: A \iff B in Fassume A1:
A \iff B in F
; :: thesis: (\not A) \iff (\not B) 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 & (A \imp B) \imp ((\not B) \imp (\not A)) in F & (B \imp A) \imp ((\not A) \imp (\not B)) in F ) by A1, Def38, Th57;

then ( (\not A) \imp (\not B) in F & (\not B) \imp (\not A) in F ) by Def38;

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

hence (\not A) \iff (\not B) in F by Def38; :: thesis: verum

end;( (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 & (A \imp B) \imp ((\not B) \imp (\not A)) in F & (B \imp A) \imp ((\not A) \imp (\not B)) in F ) by A1, Def38, Th57;

then ( (\not A) \imp (\not B) in F & (\not B) \imp (\not A) in F ) by Def38;

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

hence (\not A) \iff (\not B) in F by Def38; :: thesis: verum

( ((\not A) \iff (\not B)) \imp ((\not A) \imp (\not B)) in F & ((\not A) \iff (\not B)) \imp ((\not B) \imp (\not A)) in F ) by Def38;

then ( (\not A) \imp (\not B) in F & (\not B) \imp (\not A) in F & ((\not A) \imp (\not B)) \imp (B \imp A) in F & ((\not B) \imp (\not A)) \imp (A \imp B) in F ) by A2, Def38;

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

then ( (A \imp B) \and (B \imp A) in F & ((A \imp B) \and (B \imp A)) \imp (A \iff B) in F ) by Def38, Th35;

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