Theorem sbco4lem

Description: Lemma for sbco4 . It replaces the temporary variable v with another temporary variable w . (Contributed by Jim Kingdon, 26-Sep-2018) (Proof shortened by Wolf Lammen, 12-Oct-2024) Avoid ax-11 . (Revised by SN, 3-Sep-2025)

	Ref	Expression
Assertion	sbco4lem	⊢ ( [ 𝑥 / 𝑣 ] [ 𝑦 / 𝑥 ] [ 𝑣 / 𝑦 ] 𝜑 ↔ [ 𝑥 / 𝑤 ] [ 𝑦 / 𝑥 ] [ 𝑤 / 𝑦 ] 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		sbequ	⊢ ( 𝑣 = 𝑤 → ( [ 𝑣 / 𝑦 ] 𝜑 ↔ [ 𝑤 / 𝑦 ] 𝜑 ) )
2	1	sbbidv	⊢ ( 𝑣 = 𝑤 → ( [ 𝑦 / 𝑥 ] [ 𝑣 / 𝑦 ] 𝜑 ↔ [ 𝑦 / 𝑥 ] [ 𝑤 / 𝑦 ] 𝜑 ) )
3	2	cbvsbv	⊢ ( [ 𝑥 / 𝑣 ] [ 𝑦 / 𝑥 ] [ 𝑣 / 𝑦 ] 𝜑 ↔ [ 𝑥 / 𝑤 ] [ 𝑦 / 𝑥 ] [ 𝑤 / 𝑦 ] 𝜑 )