Proof of proveit.logic.booleans.disjunction.or_if_both theorem

import proveit
from proveit import defaults
from proveit import A, B
from proveit.logic.booleans.disjunction import true_or_true
theory = proveit.Theory() # the theorem's theory

%proving or_if_both

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

defaults.assumptions = or_if_both.all_conditions()

defaults.assumptions:

AeqT = A.evaluation()

AeqT:  ⊢  

BeqT = B.evaluation()

BeqT:  ⊢  

true_or_true

 ⊢  

AorT = AeqT.sub_left_side_into(true_or_true.inner_expr().operands[0],
                               auto_simplify=False)

AorT:  ⊢  

AorB = BeqT.sub_left_side_into(AorT, auto_simplify=False)

AorB: ,  ⊢  

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

%qed

proveit.logic.booleans.disjunction.or_if_both has been proven.

	step type	requirements	statement
0	generalization	1	⊢
1	instantiation	4, 2, 3	,  ⊢
	: , :
2	instantiation	4, 5, 6	⊢
	: , :
3	assumption		⊢
4	theorem		⊢
	proveit.logic.equality.substitute_truth
5	theorem		⊢
	proveit.logic.booleans.disjunction.true_or_true
6	assumption		⊢