Theorem zeo4 16373

Description: An integer is even or odd but not both. With this representation of even and odd integers, this variant of zeo2 12661 follows immediately from the principle of double negation, see notnotb 317. (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 317	. 2 ⊢ (2 ∥ 𝑁 ↔ ¬ ¬ 2 ∥ 𝑁)
2	1	a1i 11	1 ⊢ (𝑁 ∈ ℤ → (2 ∥ 𝑁 ↔ ¬ ¬ 2 ∥ 𝑁))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∈ wcel 2143   class class class wbr 5101  2c2 12273  ℤcz 12569   ∥ cdvds 16287

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 209

This theorem is referenced by: (None)