Proof of proveit.logic.booleans.in_bool_is_bool theorem

import proveit
from proveit import l, x, A
from proveit.numbers import two
from proveit.logic import bools_def, TRUE, FALSE
from proveit.logic.booleans import in_bool_def
from proveit.logic.sets.enumeration  import enum_set_def
theory = proveit.Theory() # the theorem's theory

%proving in_bool_is_bool

With these allowed/disallowed theorem/theory presumptions (e.g., to avoid circular dependencies), we begin our proof of
in_bool_is_bool:
(see dependencies)

in_bool_is_bool may now be readily provable (assuming required theorems are usable).  Simply execute "%qed".

in_bool_def

 ⊢  

in_bool_def_spec = in_bool_def.instantiate({A:A})

in_bool_def_spec:  ⊢  

in_bool_def_r_h_s_in_bool = in_bool_def_spec.rhs.deduce_in_bool()

in_bool_def_r_h_s_in_bool:  ⊢  

in_bool_def_spec.sub_left_side_into(in_bool_def_r_h_s_in_bool)

 ⊢  

%qed

proveit.logic.booleans.in_bool_is_bool has been proven.

	step type	requirements	statement
0	generalization	1	⊢
1	instantiation	2, 3, 4	⊢
	: , : , :
2	theorem		⊢
	proveit.logic.equality.sub_left_side_into
3	instantiation	5, 6, 7	⊢
	: , :
4	instantiation	8	⊢
	:
5	conjecture		⊢
	proveit.logic.booleans.disjunction.binary_closure
6	instantiation	9	⊢
	: , :
7	instantiation	9	⊢
	: , :
8	conjecture		⊢
	proveit.logic.booleans.in_bool_def
9	axiom		⊢
	proveit.logic.equality.equality_in_bool