let x be Variable; :: thesis: for M being non empty set
for m being Element of M
for H being ZF-formula
for v being Function of VAR ,M st not x in variables_in H holds
( M,v |= H iff M,v / x,m |= H )
let M be non empty set ; :: thesis: for m being Element of M
for H being ZF-formula
for v being Function of VAR ,M st not x in variables_in H holds
( M,v |= H iff M,v / x,m |= H )
let m be Element of M; :: thesis: for H being ZF-formula
for v being Function of VAR ,M st not x in variables_in H holds
( M,v |= H iff M,v / x,m |= H )
let H be ZF-formula; :: thesis: for v being Function of VAR ,M st not x in variables_in H holds
( M,v |= H iff M,v / x,m |= H )
let v be Function of VAR ,M; :: thesis: ( not x in variables_in H implies ( M,v |= H iff M,v / x,m |= H ) )
A1: ( M,v / x,m |= All x,H implies M,(v / x,m) / x,(v . x) |= H ) by ZF_LANG1:80;
A2: (v / x,m) / x,(v . x) = v / x,(v . x) by FUNCT_7:36;
( M,v |= All x,H implies M,v / x,m |= H ) by ZF_LANG1:80;
hence ( not x in variables_in H implies ( M,v |= H iff M,v / x,m |= H ) ) by A1, A2, Th5, FUNCT_7:37; :: thesis: verum