<s>
In	O
proof	O
theory	O
and	O
constructive	O
mathematics	O
,	O
the	O
principle	O
of	O
independence	B-Algorithm
of	I-Algorithm
premise	I-Algorithm
states	O
that	O
if	O
φ	O
and	O
∃	O
x	O
θ	O
are	O
sentences	O
in	O
a	O
formal	O
theory	O
and	O
is	O
provable	O
,	O
then	O
is	O
provable	O
.	O
</s>
<s>
The	O
principle	O
of	O
independence	B-Algorithm
of	I-Algorithm
premise	I-Algorithm
is	O
valid	O
in	O
classical	O
logic	O
because	O
of	O
the	O
law	O
of	O
the	O
excluded	O
middle	O
.	O
</s>
<s>
The	O
principle	O
of	O
independence	B-Algorithm
of	I-Algorithm
premise	I-Algorithm
is	O
not	O
generally	O
valid	O
in	O
intuitionistic	O
logic	O
(	O
Avigad	O
and	O
Feferman	O
1999	O
)	O
.	O
</s>
<s>
This	O
can	O
be	O
illustrated	O
by	O
the	O
BHK	B-Application
interpretation	I-Application
,	O
which	O
says	O
that	O
in	O
order	O
to	O
prove	O
intuitionistically	O
,	O
one	O
must	O
create	O
a	O
function	O
that	O
takes	O
a	O
proof	O
of	O
φ	O
and	O
returns	O
a	O
proof	O
of	O
.	O
</s>
