Proof of proveit.logic.equality.not_equals_is_bool theorem

import proveit
from proveit import x, y
from proveit.logic import Equals, NotEquals, Not, in_bool
from proveit.logic.equality  import not_equals_def
from proveit import x, y
theory = proveit.Theory() # the theorem's theory

%proving not_equals_is_bool

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

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

not_equals_def

 ⊢  

not_equals_def_spec = not_equals_def.instantiate({x:x, y:y})

not_equals_def_spec:  ⊢  

neg_equals_in_bool = in_bool(Not(Equals(x, y))).prove()

neg_equals_in_bool:  ⊢  

not_equals_def_spec.sub_left_side_into(neg_equals_in_bool)

 ⊢  

%qed

proveit.logic.equality.not_equals_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	⊢
	:
4	instantiation	7	⊢
	: , :
5	conjecture		⊢
	proveit.logic.booleans.negation.closure
6	instantiation	8	⊢
	: , :
7	axiom		⊢
	proveit.logic.equality.not_equals_def
8	axiom		⊢
	proveit.logic.equality.equality_in_bool