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Metamath Proof Explorer

Theorem enen1

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

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

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