Theorem sb5OLD

Description: Obsolete version of sb5 as of 21-Sep-2024. (Contributed by NM, 18-Aug-1993) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	sb5OLD	⊢ ( [ 𝑦 / 𝑥 ] 𝜑 ↔ ∃ 𝑥 ( 𝑥 = 𝑦 ∧ 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		nfs1v	⊢ Ⅎ 𝑥 [ 𝑦 / 𝑥 ] 𝜑
2		sbequ12	⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ [ 𝑦 / 𝑥 ] 𝜑 ) )
3	1 2	equsexv	⊢ ( ∃ 𝑥 ( 𝑥 = 𝑦 ∧ 𝜑 ) ↔ [ 𝑦 / 𝑥 ] 𝜑 )
4	3	bicomi	⊢ ( [ 𝑦 / 𝑥 ] 𝜑 ↔ ∃ 𝑥 ( 𝑥 = 𝑦 ∧ 𝜑 ) )