Theorem sbco2v

Description: A composition law for substitution. Version of sbco2 with disjoint variable conditions, not requiring ax-13 , but ax-11 . (Contributed by NM, 30-Jun-1994) (Revised by Wolf Lammen, 29-Apr-2023)

	Ref	Expression
Hypothesis	sbco2v.1	⊢ Ⅎ 𝑧 𝜑
Assertion	sbco2v	⊢ ( [ 𝑦 / 𝑧 ] [ 𝑧 / 𝑥 ] 𝜑 ↔ [ 𝑦 / 𝑥 ] 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		sbco2v.1	⊢ Ⅎ 𝑧 𝜑
2	1	nfsbv	⊢ Ⅎ 𝑧 [ 𝑦 / 𝑥 ] 𝜑
3		sbequ	⊢ ( 𝑧 = 𝑦 → ( [ 𝑧 / 𝑥 ] 𝜑 ↔ [ 𝑦 / 𝑥 ] 𝜑 ) )
4	2 3	sbiev	⊢ ( [ 𝑦 / 𝑧 ] [ 𝑧 / 𝑥 ] 𝜑 ↔ [ 𝑦 / 𝑥 ] 𝜑 )