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 ( A \imp B in F & B \imp 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 \iff B in F iff ( A \imp B in F & B \imp 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 \iff B in F iff ( A \imp B in F & B \imp A in F ) )

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

( A \iff B in F iff ( A \imp B in F & B \imp A in F ) )

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

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

hence ( A \iff B in F implies ( A \imp B in F & B \imp A in F ) ) by Def38; :: thesis: ( A \imp B in F & B \imp A in F implies A \iff B in F )

assume ( A \imp B in F & B \imp A in F ) ; :: thesis: A \iff B in F

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 ( A \imp B in F & B \imp 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 \iff B in F iff ( A \imp B in F & B \imp 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 \iff B in F iff ( A \imp B in F & B \imp A in F ) )

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

( A \iff B in F iff ( A \imp B in F & B \imp A in F ) )

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

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

hence ( A \iff B in F implies ( A \imp B in F & B \imp A in F ) ) by Def38; :: thesis: ( A \imp B in F & B \imp A in F implies A \iff B in F )

assume ( A \imp B in F & B \imp A in F ) ; :: thesis: A \iff B in F

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