Proof of proveit.logic.equality.substitute_falsehood theorem

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

%proving substitute_falsehood

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

defaults.assumptions = substitute_falsehood.all_conditions()

defaults.assumptions:

x_eq_F = x.evaluation()

x_eq_F:  ⊢  

x_eq_F.sub_left_side_into(PofFalse)

,  ⊢  

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

%qed

proveit.logic.equality.substitute_falsehood 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.negation.negation_elim
6	assumption		⊢