1
Q
Assume ∀x ∀y P(x,y)
A
Assumption
2
Q
∀y P(x,y)
A
Universal Instantiation
3
Q
P(x,x)
A
Universal Instantiation
4
Q
∀x P(x,x)
A
Universal Generalization
5
Q
∴ ∀x ∀y P(x,y) → ∀x P(x,x)
A
Derivation
6
Q
Assume ∀x ¬P(x)
A
Assumption
7
Q
Assume ∃x P(x)
A
Assumption
8
Q
P(x)
A
Existential Instantiation
9
Q
∀x ¬P(x)
A
Copy
10
Q
¬P(x)
A
Universal Instantiation
11
Q
P(x)
A
Copy
12
Q
P(x) ∧ ¬P(x)
A
Conjunction
13
Q
0…p ∧ ¬p ≡ 0…∴ ∃x P(x) → 0
A
Derivation
14
Q
¬∃x P(x)
A
Contradiction
15
Q
∴ ∀x ¬P(x) → ¬∃x P(x)
A
Derivation
16
Q
Assume ¬∃x P(x)
A
Assumption
17
Q
Let x be arbitrary…Comment….Assume P(x)
A
Assumption
18
Q
∃x P(x)
A
Existential Generalization
19
Q
¬∃x P(x)
A
Copy
20
Q
∃x P(x)..∧ ¬∃x P(x)
A
Conjunction
21
Q
0…p ∧ ¬p ≡ 0…∴ P(x) → 0
A
Derivation
22
Q
¬P(x)
A
Contradiction
23
Q
∀x ¬P(x)
A
Universal Generalization
24
Q
∴ ¬∃x P(x) → ∀x ¬P(x)
A
Derivation
25
∀x ¬P(x) → ¬∃x P(x)
Copy
26
(¬∃x P(x) → ∀x ¬P(x)) ∧ (∀x ¬P(x) → ¬∃x P(x))
Conjunction
27
(¬∃x P(x) ↔ ∀x ¬P(x))...(a ↔ b) ≡ (a → b) ∧ (b → a)
| ∀P (¬∃x P(x) ↔ ∀x ¬P(x)).
Universal Generalization
28
Assume ∀x P(x)
Assumption
29
P(0)
Universal Instantiation
30
∃x P(x)
Existential Generalization
31
∴ ∀x P(x) → ∃x P(x)
Derivation
Math Concepts In Computing
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