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

import proveit
# we need to import true_iff_false_negated and false_iff_true_negated
# to derive their side-effects:
from proveit.logic.booleans.implication import true_iff_true, true_iff_false_negated
from proveit.logic.booleans.implication import false_iff_false, false_iff_true_negated
theory = proveit.Theory() # the theorem's theory

%proving iff_closure

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

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

%qed

proveit.logic.booleans.implication.iff_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	18, 12	⊢
	:
8	instantiation	14, 13	⊢
	:
9	conjecture		⊢
	proveit.logic.booleans.forall_over_bool_by_cases
10	instantiation	14, 15	⊢
	:
11	instantiation	18, 16	⊢
	:
12	conjecture		⊢
	proveit.logic.booleans.implication.true_iff_true
13	instantiation	18, 17	⊢
	:
14	axiom		⊢
	proveit.logic.booleans.negation.operand_is_bool
15	instantiation	18, 19	⊢
	:
16	conjecture		⊢
	proveit.logic.booleans.implication.false_iff_false
17	theorem		⊢
	proveit.logic.booleans.implication.true_iff_false_negated
18	conjecture		⊢
	proveit.logic.booleans.in_bool_if_true
19	theorem		⊢
	proveit.logic.booleans.implication.false_iff_true_negated