Proof of proveit.logic.equality.sub_left_side_into theorem

import proveit
from proveit import defaults
from proveit import f, x, y, P, Px, Py
from proveit.logic import Equals
from proveit.logic.equality  import substitution
theory = proveit.Theory() # the theorem's theory

%proving sub_left_side_into

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

defaults.assumptions = sub_left_side_into.all_conditions()

defaults.assumptions:

substitution

 ⊢  

# P(x)=P(y) from x=y via substitution
substitution.instantiate({f:P, x:x, y:y})

 ⊢  

Py.prove().evaluation()

 ⊢  

# P(x)=TRUE via P(y)=TRUE and transitivity
Px.evaluation()

,  ⊢  

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

%qed

proveit.logic.equality.sub_left_side_into has been proven.

	step type	requirements	statement
0	generalization	1	⊢
1	instantiation	2, 3	,  ⊢
	:
2	axiom		⊢
	proveit.logic.booleans.eq_true_elim
3	instantiation	4, 5, 6	,  ⊢
	: , : , :
4	axiom		⊢
	proveit.logic.equality.equals_transitivity
5	instantiation	7, 8	⊢
	: , : , :
6	instantiation	9, 10	⊢
	:
7	axiom		⊢
	proveit.logic.equality.substitution
8	assumption		⊢
9	axiom		⊢
	proveit.logic.booleans.eq_true_intro
10	assumption		⊢