Proof of proveit.logic.booleans.implication.implication_closure theorem

import proveit
# we need to import true_implies_false_negated to derive it's side-effects:
from proveit.logic.booleans.implication import true_implies_false_negated
theory = proveit.Theory() # the theorem's theory

%proving implication_closure

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

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

%qed

proveit.logic.booleans.implication.implication_closure has been proven.

	step type	requirements	statement
0	modus ponens	1, 2	⊢
1	instantiation	3, 4	⊢
	: , : , : , : , : , : , :
2	instantiation	9, 5, 6	⊢
	: , :
3	conjecture		⊢
	proveit.logic.booleans.quantification.universality.bundle
4	conjecture		⊢
	proveit.numbers.numerals.decimals.posnat1
5	instantiation	9, 7, 8	⊢
	: , :
6	instantiation	9, 10, 11	⊢
	: , :
7	instantiation	17, 12	⊢
	:
8	instantiation	13, 14	⊢
	:
9	conjecture		⊢
	proveit.logic.booleans.forall_over_bool_by_cases
10	instantiation	17, 15	⊢
	:
11	instantiation	17, 16	⊢
	:
12	theorem		⊢
	proveit.logic.booleans.implication.true_implies_true
13	axiom		⊢
	proveit.logic.booleans.negation.operand_is_bool
14	instantiation	17, 18	⊢
	:
15	theorem		⊢
	proveit.logic.booleans.implication.false_implies_true
16	theorem		⊢
	proveit.logic.booleans.implication.false_implies_false
17	conjecture		⊢
	proveit.logic.booleans.in_bool_if_true
18	theorem		⊢
	proveit.logic.booleans.implication.true_implies_false_negated