let p, q, r be ZF-formula; :: thesis: for M being non empty set
for v being Function of VAR,M holds
( M,v |= (p => (q => r)) => ((p => q) => (p => r)) & M |= (p => (q => r)) => ((p => q) => (p => r)) )
let M be non empty set ; :: thesis: for v being Function of VAR,M holds
( M,v |= (p => (q => r)) => ((p => q) => (p => r)) & M |= (p => (q => r)) => ((p => q) => (p => r)) )
let v be Function of VAR,M; :: thesis: ( M,v |= (p => (q => r)) => ((p => q) => (p => r)) & M |= (p => (q => r)) => ((p => q) => (p => r)) )

now :: thesis: for v being Function of VAR,M holds M,v |= (p => (q => r)) => ((p => q) => (p => r))

let v be Function of VAR,M; :: thesis: M,v |= (p => (q => r)) => ((p => q) => (p => r))

now :: thesis: ( M,v |= p => (q => r) implies M,v |= (p => q) => (p => r) )

assume A1: M,v |= p => (q => r) ; :: thesis: M,v |= (p => q) => (p => r)

now :: thesis: ( M,v |= p => q implies M,v |= p => r )

assume M,v |= p => q ; :: thesis: M,v |= p => r
then A2: ( M,v |= p implies ( M,v |= q => r & M,v |= q ) ) by A1, ZF_MODEL:18;
( M,v |= q & M,v |= q => r implies M,v |= r ) by ZF_MODEL:18;
hence M,v |= p => r by A2, ZF_MODEL:18; :: thesis: verum

end;

hence M,v |= (p => q) => (p => r) by ZF_MODEL:18; :: thesis: verum

end;

hence M,v |= (p => (q => r)) => ((p => q) => (p => r)) by ZF_MODEL:18; :: thesis: verum

end;

hence ( M,v |= (p => (q => r)) => ((p => q) => (p => r)) & M |= (p => (q => r)) => ((p => q) => (p => r)) ) ; :: thesis: verum