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Metamath Proof Explorer

Theorem equequ2

Description: An equivalence law for equality. (Contributed by NM, 21-Jun-1993) (Proof shortened by Wolf Lammen, 4-Aug-2017) (Proof shortened by BJ, 12-Apr-2021)

		Ref	Expression
	Assertion	equequ2	⊢ ( 𝑥 = 𝑦 → ( 𝑧 = 𝑥 ↔ 𝑧 = 𝑦 ) )

Proof

Step	Hyp	Ref	Expression
1		equtrr	⊢ ( 𝑥 = 𝑦 → ( 𝑧 = 𝑥 → 𝑧 = 𝑦 ) )
2		equeuclr	⊢ ( 𝑥 = 𝑦 → ( 𝑧 = 𝑦 → 𝑧 = 𝑥 ) )
3	1 2	impbid	⊢ ( 𝑥 = 𝑦 → ( 𝑧 = 𝑥 ↔ 𝑧 = 𝑦 ) )