Proof of proveit.logic.equality.substitute_truth theorem

import proveit
from proveit import defaults
from proveit import x, P, Px
from proveit.logic import PofTrue
theory = proveit.Theory() # the theorem's theory

%proving substitute_truth

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

defaults.assumptions = substitute_truth.all_conditions()

defaults.assumptions:

x_eq_T = x.evaluation()

x_eq_T:  ⊢  

x_eq_T.sub_left_side_into(PofTrue)

,  ⊢  

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

%qed

proveit.logic.equality.substitute_truth has been proven.

	step type	requirements	statement
0	generalization	1	⊢
1	instantiation	2, 3, 4	,  ⊢
	: , : , :
2	conjecture		⊢
	proveit.logic.equality.sub_left_side_into
3	assumption		⊢
4	instantiation	5, 6	⊢
	:
5	axiom		⊢
	proveit.logic.booleans.eq_true_intro
6	assumption		⊢