Theorem enen1

Description: Equality-like theorem for equinumerosity. (Contributed by NM, 18-Dec-2003)

	Ref	Expression
Assertion	enen1	$$⊢A\approx B\to \left(A\approx C\leftrightarrow B\approx C\right)$$

Proof

Step	Hyp	Ref	Expression
1		ensym	$$⊢A\approx B\to B\approx A$$
2		entr	$$⊢\left(B\approx A\wedge A\approx C\right)\to B\approx C$$
3	1 2	sylan	$$⊢\left(A\approx B\wedge A\approx C\right)\to B\approx C$$
4		entr	$$⊢\left(A\approx B\wedge B\approx C\right)\to A\approx C$$
5	3 4	impbida	$$⊢A\approx B\to \left(A\approx C\leftrightarrow B\approx C\right)$$