Proof of proveit.logic.booleans.in_bool_def theorem

import proveit
from proveit import l, x, A
from proveit.logic import bools_def, TRUE, FALSE, InSet
theory = proveit.Theory() # the theorem's theory

%proving in_bool_def

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

bools_def

 ⊢  

Ain_boolSet = InSet(A, bools_def.rhs)

Ain_boolSet:

in_bool_set_def = Ain_boolSet.definition()

in_bool_set_def:  ⊢  

bools_def.sub_left_side_into(in_bool_set_def)

 ⊢  

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

%qed

proveit.logic.booleans.in_bool_def 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	axiom		⊢
	proveit.logic.booleans.bools_def
5	axiom		⊢
	proveit.logic.sets.enumeration.enum_set_def
6	conjecture		⊢
	proveit.numbers.numerals.decimals.nat2
7	instantiation	8	⊢
	: , :
8	conjecture		⊢
	proveit.numbers.numerals.decimals.tuple_len_2_typical_eq