let E be non empty set ; :: thesis: for f being Function of VAR ,E
for H, H' being ZF-formula holds
( E,f |= H <=> H' iff ( E,f |= H iff E,f |= H' ) )
let f be Function of VAR ,E; :: thesis: for H, H' being ZF-formula holds
( E,f |= H <=> H' iff ( E,f |= H iff E,f |= H' ) )
let H, H' be ZF-formula; :: thesis: ( E,f |= H <=> H' iff ( E,f |= H iff E,f |= H' ) )
( E,f |= (H => H') '&' (H' => H) iff ( E,f |= H => H' & E,f |= H' => H ) ) by Th15;
hence ( E,f |= H <=> H' iff ( E,f |= H iff E,f |= H' ) ) by Th18; :: thesis: verum