Theorem exists_def_negation of type Forall

from the theory of proveit.logic.booleans.quantification.existence

see dependencies

import proveit
# Automation is not needed when only building an expression:
proveit.defaults.automation = False # This will speed things up.
proveit.defaults.inline_pngs = False # Makes files smaller.
%load_theorem_expr # Load the stored theorem expression as 'stored_expr'
# import the special expression
from proveit.logic.booleans.quantification.existence import exists_def_negation

# check that the built expression is the same as the stored expression
assert exists_def_negation.expr == stored_expr
assert exists_def_negation.expr._style_id == stored_expr._style_id
print("Passed sanity check: exists_def_negation matches stored_expr")

Passed sanity check: exists_def_negation matches stored_expr

# Show the LaTeX representation of the expression for convenience if you need it.
print(stored_expr.latex())

\forall_{n \in \mathbb{N}^+}~\left[\forall_{P, Q}~\left(\begin{array}{c} \begin{array}{l} \left[\nexists_{x_{1}, x_{2}, \ldots, x_{n}~|~Q\left(x_{1}, x_{2}, \ldots, x_{n}\right)}~P\left(x_{1}, x_{2}, \ldots, x_{n}\right)\right] =  \\ \left[\forall_{y_{1}, y_{2}, \ldots, y_{n}~|~Q\left(y_{1}, y_{2}, \ldots, y_{n}\right)}~\left(P\left(y_{1}, y_{2}, \ldots, y_{n}\right) \neq \top\right)\right] \end{array} \end{array}\right)\right]

stored_expr.style_options()

name	description	default	current value	related methods
with_wrapping	If 'True', wrap the Expression after the parameters	None	None/False	('with_wrapping',)
condition_wrapping	Wrap 'before' or 'after' the condition (or None).	None	None/False	('with_wrap_after_condition', 'with_wrap_before_condition')
wrap_params	If 'True', wraps every two parameters AND wraps the Expression after the parameters	None	None/False	('with_params',)
justification	justify to the 'left', 'center', or 'right' in the array cells	center	center	('with_justification',)

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Operation	operator: 20 operand: 2
1	ExprTuple	2
2	Lambda	parameter: 45 body: 4
3	ExprTuple	45
4	Conditional	value: 5 condition: 6
5	Operation	operator: 20 operand: 10
6	Operation	operator: 8 operands: 9
7	ExprTuple	10
8	Literal
9	ExprTuple	45, 11
10	Lambda	parameters: 12 body: 13
11	Literal
12	ExprTuple	38, 33
13	Operation	operator: 14 operands: 15
14	Literal
15	ExprTuple	16, 17
16	Operation	operator: 18 operand: 22
17	Operation	operator: 20 operand: 23
18	Literal
19	ExprTuple	22
20	Literal
21	ExprTuple	23
22	Lambda	parameters: 30 body: 24
23	Lambda	parameters: 39 body: 25
24	Conditional	value: 26 condition: 27
25	Conditional	value: 28 condition: 29
26	Operation	operator: 38 operands: 30
27	Operation	operator: 33 operands: 30
28	Operation	operator: 31 operands: 32
29	Operation	operator: 33 operands: 39
30	ExprTuple	34
31	Literal
32	ExprTuple	35, 36
33	Variable
34	ExprRange	lambda_map: 37 start_index: 44 end_index: 45
35	Operation	operator: 38 operands: 39
36	Literal
37	Lambda	parameter: 49 body: 40
38	Variable
39	ExprTuple	41
40	IndexedVar	variable: 42 index: 49
41	ExprRange	lambda_map: 43 start_index: 44 end_index: 45
42	Variable
43	Lambda	parameter: 49 body: 46
44	Literal
45	Variable
46	IndexedVar	variable: 47 index: 49
47	Variable
48	ExprTuple	49
49	Variable