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Metamath Proof Explorer

Theorem equsb1v

Description: Substitution applied to an atomic wff. Version of equsb1 with a disjoint variable condition, which neither requires ax-12 nor ax-13 . (Contributed by NM, 10-May-1993) (Revised by BJ, 11-Sep-2019) Remove dependencies on axioms. (Revised by Wolf Lammen, 30-May-2023) (Proof shortened by Steven Nguyen, 19-Jun-2023) Revise df-sb . (Revised by Steven Nguyen, 11-Jul-2023) (Proof shortened by Steven Nguyen, 22-Jul-2023)

		Ref	Expression
	Assertion	equsb1v	⊢ [ 𝑦 / 𝑥 ] 𝑥 = 𝑦

Proof

Step	Hyp	Ref	Expression
1		equid	⊢ 𝑦 = 𝑦
2		equsb3	⊢ ( [ 𝑦 / 𝑥 ] 𝑥 = 𝑦 ↔ 𝑦 = 𝑦 )
3	1 2	mpbir	⊢ [ 𝑦 / 𝑥 ] 𝑥 = 𝑦