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Metamath Proof Explorer

Theorem equtr2

Description: Equality is a left-Euclidean binary relation. Uncurried (imported) form of equeucl . (Contributed by NM, 12-Aug-1993) (Proof shortened by Andrew Salmon, 25-May-2011) (Proof shortened by BJ, 11-Apr-2021)

		Ref	Expression
	Assertion	equtr2	⊢ ( ( 𝑥 = 𝑧 ∧ 𝑦 = 𝑧 ) → 𝑥 = 𝑦 )

Proof

Step	Hyp	Ref	Expression
1		equeucl	⊢ ( 𝑥 = 𝑧 → ( 𝑦 = 𝑧 → 𝑥 = 𝑦 ) )
2	1	imp	⊢ ( ( 𝑥 = 𝑧 ∧ 𝑦 = 𝑧 ) → 𝑥 = 𝑦 )