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 being Formula of L st C \imp (B \imp A) in F & B in F holds

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

C \imp A in F

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

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

C \imp A in F

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

C \imp A in F

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

assume that

A1: C \imp (B \imp A) in F and

A2: B in F ; :: thesis: C \imp A in F

(C \imp (B \imp A)) \imp (B \imp (C \imp A)) in F by Th41;

then B \imp (C \imp A) in F by A1, Def38;

hence C \imp A in F by A2, Def38; :: thesis: verum

for L being language MSAlgebra over S

for F being PC-theory of L

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

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

C \imp A in F

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

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

C \imp A in F

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

C \imp A in F

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

assume that

A1: C \imp (B \imp A) in F and

A2: B in F ; :: thesis: C \imp A in F

(C \imp (B \imp A)) \imp (B \imp (C \imp A)) in F by Th41;

then B \imp (C \imp A) in F by A1, Def38;

hence C \imp A in F by A2, Def38; :: thesis: verum