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

import proveit
from proveit import defaults
from proveit import A, B
from proveit.logic import And, Implies, Iff, Not
from proveit.logic.booleans.conjunction import true_and_false_negated
from proveit.logic.booleans.implication import negated_reflex
theory = proveit.Theory() # the theorem's theory

%proving not_iff_via_not_left_impl

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

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

defaults.assumptions = not_iff_via_not_left_impl.all_conditions()

defaults.assumptions:

true_and_false_negated

 ⊢  

not__AimplB_and_T = true_and_false_negated.inner_expr().operand.operands[0].substitute(Implies(A, B))

not__AimplB_and_T:  ⊢  

not__AimplB_and_BimplA = not__AimplB_and_T.inner_expr().operand.operands[1].substitute(Implies(B, A))

not__AimplB_and_BimplA:  ⊢  

A_iff_B__def = Iff(A, B).definition(assumptions=[])

A_iff_B__def:  ⊢  

A_iff_B__def.sub_left_side_into(not__AimplB_and_BimplA)

 ⊢  

%qed

proveit.logic.booleans.implication.not_iff_via_not_left_impl 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, 12	⊢
	: , :
4	instantiation	7	⊢
	: , :
5	theorem		⊢
	proveit.logic.equality.substitute_falsehood
6	instantiation	8, 9, 10	⊢
	: , :
7	axiom		⊢
	proveit.logic.booleans.implication.iff_def
8	theorem		⊢
	proveit.logic.equality.substitute_truth
9	theorem		⊢
	proveit.logic.booleans.conjunction.true_and_false_negated
10	instantiation	11, 12	⊢
	: , :
11	theorem		⊢
	proveit.logic.booleans.implication.negated_reflex
12	assumption		⊢