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

now

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

now

assume M,v |= (p => q) '&' (q => r) ; :: thesis: M,v |= p => r
then ( M,v |= p => q & M,v |= q => r ) by ZF_MODEL:15;
hence M,v |= p => r by Th113; :: thesis: verum

end;

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

end;

hence ( M,v |= ((p => q) '&' (q => r)) => (p => r) & M |= ((p => q) '&' (q => r)) => (p => r) ) by ZF_MODEL:def 5; :: thesis: verum