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, C, D being Formula of L st A \iff (B \and C) in F & C \iff D in F holds

A \iff (B \and D) 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, C, D being Formula of L st A \iff (B \and C) in F & C \iff D in F holds

A \iff (B \and D) in F

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

for A, B, C, D being Formula of L st A \iff (B \and C) in F & C \iff D in F holds

A \iff (B \and D) in F

let F be PC-theory of L; :: thesis: for A, B, C, D being Formula of L st A \iff (B \and C) in F & C \iff D in F holds

A \iff (B \and D) in F

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

assume A1: A \iff (B \and C) in F ; :: thesis: ( not C \iff D in F or A \iff (B \and D) in F )

then A2: (B \and C) \iff A in F by Th90;

assume C \iff D in F ; :: thesis: A \iff (B \and D) in F

then ( C \imp D in F & D \imp C in F & B \imp B in F ) by Th43, Th34;

then ( (B \and C) \imp (B \and D) in F & (B \and D) \imp (B \and C) in F ) by Th72;

then ( A \imp (B \and D) in F & (B \and D) \imp A in F ) by A1, A2, Th92, Th93;

hence A \iff (B \and D) in F by Th43; :: thesis: verum

for L being language MSAlgebra over S

for F being PC-theory of L

for A, B, C, D being Formula of L st A \iff (B \and C) in F & C \iff D in F holds

A \iff (B \and D) 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, C, D being Formula of L st A \iff (B \and C) in F & C \iff D in F holds

A \iff (B \and D) in F

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

for A, B, C, D being Formula of L st A \iff (B \and C) in F & C \iff D in F holds

A \iff (B \and D) in F

let F be PC-theory of L; :: thesis: for A, B, C, D being Formula of L st A \iff (B \and C) in F & C \iff D in F holds

A \iff (B \and D) in F

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

assume A1: A \iff (B \and C) in F ; :: thesis: ( not C \iff D in F or A \iff (B \and D) in F )

then A2: (B \and C) \iff A in F by Th90;

assume C \iff D in F ; :: thesis: A \iff (B \and D) in F

then ( C \imp D in F & D \imp C in F & B \imp B in F ) by Th43, Th34;

then ( (B \and C) \imp (B \and D) in F & (B \and D) \imp (B \and C) in F ) by Th72;

then ( A \imp (B \and D) in F & (B \and D) \imp A in F ) by A1, A2, Th92, Th93;

hence A \iff (B \and D) in F by Th43; :: thesis: verum