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Metamath Proof Explorer

Theorem equeucl

Description: Equality is a left-Euclidean binary relation. (Right-Euclideanness is stated in ax-7 .) Curried (exported) form of equtr2 . (Contributed by BJ, 11-Apr-2021)

		Ref	Expression
	Assertion	equeucl	⊢ ( 𝑥 = 𝑧 → ( 𝑦 = 𝑧 → 𝑥 = 𝑦 ) )

Proof

Step	Hyp	Ref	Expression
1		equeuclr	⊢ ( 𝑦 = 𝑧 → ( 𝑥 = 𝑧 → 𝑥 = 𝑦 ) )
2	1	com12	⊢ ( 𝑥 = 𝑧 → ( 𝑦 = 𝑧 → 𝑥 = 𝑦 ) )