Proof of proveit.logic.booleans.conjunction.commutation theorem

import proveit
from proveit import defaults
theory = proveit.Theory() # the theorem's theory

%proving commutation

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

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

Not quite readily provable because of unusable proofs to avoid circular logic.

defaults.assumptions = commutation.conditions

defaults.assumptions:

commutation.instance_expr.conclude_via_mutual_implication()

,  ⊢  

%qed

proveit.logic.booleans.conjunction.commutation has been proven.

	step type	requirements	statement
0	generalization	1	⊢
1	instantiation	2, 3, 4, 5, 6	,  ⊢
	: , :
2	conjecture		⊢
	proveit.logic.booleans.implication.eq_from_mutual_impl
3	instantiation	7, 9, 8	,  ⊢
	: , :
4	instantiation	7, 8, 9	,  ⊢
	: , :
5	deduction	10	⊢
6	deduction	11	⊢
7	conjecture		⊢
	proveit.logic.booleans.conjunction.binary_closure
8	assumption		⊢
9	assumption		⊢
10	instantiation	14, 12, 13	⊢
	: , :
11	instantiation	14, 15, 16	⊢
	: , :
12	instantiation	18, 17	⊢
	: , :
13	instantiation	19, 17	⊢
	: , :
14	theorem		⊢
	proveit.logic.booleans.conjunction.and_if_both
15	instantiation	18, 20	⊢
	: , :
16	instantiation	19, 20	⊢
	: , :
17	assumption		⊢
18	theorem		⊢
	proveit.logic.booleans.conjunction.right_from_and
19	theorem		⊢
	proveit.logic.booleans.conjunction.left_from_and
20	assumption		⊢