Theorem iseven2

Description: The predicate "is an even number". An even number is an integer which is divisible by 2. (Contributed by AV, 18-Jun-2020)

	Ref	Expression
Assertion	iseven2	$$⊢Z\in \mathrm{Even}\leftrightarrow \left(Z\in \mathbb{Z}\wedge 2\parallel Z\right)$$

Proof

Step	Hyp	Ref	Expression
1		breq2	$$⊢z=Z\to \left(2\parallel z\leftrightarrow 2\parallel Z\right)$$
2		dfeven2	$$⊢\mathrm{Even}=\left\{z\in \mathbb{Z}|2\parallel z\right\}$$
3	1 2	elrab2	$$⊢Z\in \mathrm{Even}\leftrightarrow \left(Z\in \mathbb{Z}\wedge 2\parallel Z\right)$$