let Al be QC-alphabet ; :: thesis: for A being non empty set
for x being bound_QC-variable of Al
for v being Element of Valuations_in (Al,A)
for p being Element of CQC-WFF Al
for J being interpretation of Al,A holds
( J,v |= All (x,p) iff (FOR_ALL (x,(Valid (p,J)))) . v = TRUE )
let A be non empty set ; :: thesis: for x being bound_QC-variable of Al
for v being Element of Valuations_in (Al,A)
for p being Element of CQC-WFF Al
for J being interpretation of Al,A holds
( J,v |= All (x,p) iff (FOR_ALL (x,(Valid (p,J)))) . v = TRUE )
let x be bound_QC-variable of Al; :: thesis: for v being Element of Valuations_in (Al,A)
for p being Element of CQC-WFF Al
for J being interpretation of Al,A holds
( J,v |= All (x,p) iff (FOR_ALL (x,(Valid (p,J)))) . v = TRUE )
let v be Element of Valuations_in (Al,A); :: thesis: for p being Element of CQC-WFF Al
for J being interpretation of Al,A holds
( J,v |= All (x,p) iff (FOR_ALL (x,(Valid (p,J)))) . v = TRUE )
let p be Element of CQC-WFF Al; :: thesis: for J being interpretation of Al,A holds
( J,v |= All (x,p) iff (FOR_ALL (x,(Valid (p,J)))) . v = TRUE )
let J be interpretation of Al,A; :: thesis: ( J,v |= All (x,p) iff (FOR_ALL (x,(Valid (p,J)))) . v = TRUE )

A1: now :: thesis: ( (FOR_ALL (x,(Valid (p,J)))) . v = TRUE implies J,v |= All (x,p) )

assume (FOR_ALL (x,(Valid (p,J)))) . v = TRUE ; :: thesis: J,v |= All (x,p)
then (Valid ((All (x,p)),J)) . v = TRUE by Lm1;
hence J,v |= All (x,p) by Def7; :: thesis: verum

end;

now :: thesis: ( J,v |= All (x,p) implies (FOR_ALL (x,(Valid (p,J)))) . v = TRUE )

assume J,v |= All (x,p) ; :: thesis: (FOR_ALL (x,(Valid (p,J)))) . v = TRUE
then (Valid ((All (x,p)),J)) . v = TRUE by Def7;
hence (FOR_ALL (x,(Valid (p,J)))) . v = TRUE by Lm1; :: thesis: verum

end;

hence ( J,v |= All (x,p) iff (FOR_ALL (x,(Valid (p,J)))) . v = TRUE ) by A1; :: thesis: verum