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Metamath Proof Explorer

Theorem equvelv

Description: A biconditional form of equvel with disjoint variable conditions and proved from Tarski's FOL axiom schemes. (Contributed by Andrew Salmon, 2-Jun-2011) Reduce axiom usage. (Revised by Wolf Lammen, 10-Apr-2021) (Proof shortened by Wolf Lammen, 12-Jul-2022)

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

Proof

Step	Hyp	Ref	Expression
1		equequ1	⊢ ( 𝑧 = 𝑥 → ( 𝑧 = 𝑦 ↔ 𝑥 = 𝑦 ) )
2	1	equsalvw	⊢ ( ∀ 𝑧 ( 𝑧 = 𝑥 → 𝑧 = 𝑦 ) ↔ 𝑥 = 𝑦 )