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

import proveit
from proveit import defaults
from proveit import A, B
from proveit.logic.booleans.negation import negation_contradiction
from proveit.logic import FALSE, Not, Equals
#from proveit.logic.booleans import from_not_false
#from proveit.logic.booleans.implication import modus_tollens_affirmation
theory = proveit.Theory() # the theorem's theory

%proving modus_tollens_denial

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

defaults.assumptions = modus_tollens_denial.all_conditions()

defaults.assumptions:

# Prove Not(A) via the contradition, under the assumptions, 
# of B and Not(B) simultaneously being true when A is true.
Not(B).deny_via_contradiction(A)

, ,  ⊢  

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

%qed

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

	step type	requirements	statement
0	generalization	1	⊢
1	instantiation	2, 3, 4	, ,  ⊢
	:
2	axiom		⊢
	proveit.logic.booleans.implication.denial_via_contradiction
3	assumption		⊢
4	deduction	5	,  ⊢
5	instantiation	6, 7, 8	, ,  ⊢
	:
6	theorem		⊢
	proveit.logic.booleans.negation.negation_contradiction
7	modus ponens	9, 10	,  ⊢
8	assumption		⊢
9	assumption		⊢
10	assumption		⊢