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 \or C) in F & B \iff D in F holds
A \iff (D \or C) 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 \or C) in F & B \iff D in F holds
A \iff (D \or C) 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 \or C) in F & B \iff D in F holds
A \iff (D \or C) in F
let F be PC-theory of L; :: thesis: for A, B, C, D being Formula of L st A \iff (B \or C) in F & B \iff D in F holds
A \iff (D \or C) in F
let A, B, C, D be Formula of L; :: thesis: ( A \iff (B \or C) in F & B \iff D in F implies A \iff (D \or C) in F )
assume A1: A \iff (B \or C) in F ; :: thesis: ( not B \iff D in F or A \iff (D \or C) in F )
then A2: (B \or C) \iff A in F by Th90;
assume B \iff D in F ; :: thesis: A \iff (D \or C) in F
then ( B \imp D in F & D \imp B in F & C \imp C in F ) by Th43, Th34;
then ( (B \or C) \imp (D \or C) in F & (D \or C) \imp (B \or C) in F ) by Th59;
then ( A \imp (D \or C) in F & (D \or C) \imp A in F ) by A1, A2, Th92, Th93;
hence A \iff (D \or C) in F by Th43; :: thesis: verum