let x be bound_QC-variable; :: thesis: for A being non empty set
for J being interpretation of A
for v being Element of Valuations_in A
for S being Element of CQC-Sub-WFF
for xSQ being second_Q_comp of [S,x] st [S,x] is quantifiable holds
( ( for a being Element of A holds J,v . ((NEx_Val v,S,x,xSQ) +* (x | a)) |= S ) iff for a being Element of A holds J,(v . (NEx_Val v,S,x,xSQ)) . (x | a) |= S )
let A be non empty set ; :: thesis: for J being interpretation of A
for v being Element of Valuations_in A
for S being Element of CQC-Sub-WFF
for xSQ being second_Q_comp of [S,x] st [S,x] is quantifiable holds
( ( for a being Element of A holds J,v . ((NEx_Val v,S,x,xSQ) +* (x | a)) |= S ) iff for a being Element of A holds J,(v . (NEx_Val v,S,x,xSQ)) . (x | a) |= S )
let J be interpretation of A; :: thesis: for v being Element of Valuations_in A
for S being Element of CQC-Sub-WFF
for xSQ being second_Q_comp of [S,x] st [S,x] is quantifiable holds
( ( for a being Element of A holds J,v . ((NEx_Val v,S,x,xSQ) +* (x | a)) |= S ) iff for a being Element of A holds J,(v . (NEx_Val v,S,x,xSQ)) . (x | a) |= S )
let v be Element of Valuations_in A; :: thesis: for S being Element of CQC-Sub-WFF
for xSQ being second_Q_comp of [S,x] st [S,x] is quantifiable holds
( ( for a being Element of A holds J,v . ((NEx_Val v,S,x,xSQ) +* (x | a)) |= S ) iff for a being Element of A holds J,(v . (NEx_Val v,S,x,xSQ)) . (x | a) |= S )
let S be Element of CQC-Sub-WFF ; :: thesis: for xSQ being second_Q_comp of [S,x] st [S,x] is quantifiable holds
( ( for a being Element of A holds J,v . ((NEx_Val v,S,x,xSQ) +* (x | a)) |= S ) iff for a being Element of A holds J,(v . (NEx_Val v,S,x,xSQ)) . (x | a) |= S )
let xSQ be second_Q_comp of [S,x]; :: thesis: ( [S,x] is quantifiable implies ( ( for a being Element of A holds J,v . ((NEx_Val v,S,x,xSQ) +* (x | a)) |= S ) iff for a being Element of A holds J,(v . (NEx_Val v,S,x,xSQ)) . (x | a) |= S ) )
assume [S,x] is quantifiable ; :: thesis: ( ( for a being Element of A holds J,v . ((NEx_Val v,S,x,xSQ) +* (x | a)) |= S ) iff for a being Element of A holds J,(v . (NEx_Val v,S,x,xSQ)) . (x | a) |= S )
thus ( ( for a being Element of A holds J,v . ((NEx_Val v,S,x,xSQ) +* (x | a)) |= S ) implies for a being Element of A holds J,(v . (NEx_Val v,S,x,xSQ)) . (x | a) |= S ) :: thesis: ( ( for a being Element of A holds J,(v . (NEx_Val v,S,x,xSQ)) . (x | a) |= S ) implies for a being Element of A holds J,v . ((NEx_Val v,S,x,xSQ) +* (x | a)) |= S )

proof

assume A1: for a being Element of A holds J,v . ((NEx_Val v,S,x,xSQ) +* (x | a)) |= S ; :: thesis: for a being Element of A holds J,(v . (NEx_Val v,S,x,xSQ)) . (x | a) |= S
let a be Element of A; :: thesis: J,(v . (NEx_Val v,S,x,xSQ)) . (x | a) |= S
v . ((NEx_Val v,S,x,xSQ) +* (x | a)) = (v . (NEx_Val v,S,x,xSQ)) . (x | a) by FUNCT_4:15;
hence J,(v . (NEx_Val v,S,x,xSQ)) . (x | a) |= S by A1; :: thesis: verum

end;

thus ( ( for a being Element of A holds J,(v . (NEx_Val v,S,x,xSQ)) . (x | a) |= S ) implies for a being Element of A holds J,v . ((NEx_Val v,S,x,xSQ) +* (x | a)) |= S ) :: thesis: verum

proof

assume A2: for a being Element of A holds J,(v . (NEx_Val v,S,x,xSQ)) . (x | a) |= S ; :: thesis: for a being Element of A holds J,v . ((NEx_Val v,S,x,xSQ) +* (x | a)) |= S
let a be Element of A; :: thesis: J,v . ((NEx_Val v,S,x,xSQ) +* (x | a)) |= S
v . ((NEx_Val v,S,x,xSQ) +* (x | a)) = (v . (NEx_Val v,S,x,xSQ)) . (x | a) by FUNCT_4:15;
hence J,v . ((NEx_Val v,S,x,xSQ) +* (x | a)) |= S by A2; :: thesis: verum

end;