Theorem zeo4 16523

Description: An integer is even or odd but not both. With this representation of even and odd integers, this variant of zeo2 12799 follows immediately from the principle of double negation, see notnotb 318. (Contributed by AV, 17-Jun-2021.)

Assertion

Ref	Expression
zeo4	⊢ (𝑁 ∈ ℤ → (2 ∥ 𝑁 ↔ ¬ ¬ 2 ∥ 𝑁))

Proof of Theorem zeo4

Step	Hyp	Ref	Expression
1		notnotb 318	. 2 ⊢ (2 ∥ 𝑁 ↔ ¬ ¬ 2 ∥ 𝑁)
2	1	a1i 11	1 ⊢ (𝑁 ∈ ℤ → (2 ∥ 𝑁 ↔ ¬ ¬ 2 ∥ 𝑁))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∈ wcel 2146   class class class wbr 5106  2c2 12410  ℤcz 12706   ∥ cdvds 16437

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210

This theorem is used by: (None)