Theorem sbcov

Description: A composition law for substitution. Version of sbco with a disjoint variable condition using fewer axioms. (Contributed by NM, 14-May-1993) (Revised by Gino Giotto, 7-Aug-2023)

	Ref	Expression
Assertion	sbcov	⊢ ( [ 𝑦 / 𝑥 ] [ 𝑥 / 𝑦 ] 𝜑 ↔ [ 𝑦 / 𝑥 ] 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		sbcom3vv	⊢ ( [ 𝑦 / 𝑥 ] [ 𝑥 / 𝑦 ] 𝜑 ↔ [ 𝑦 / 𝑥 ] [ 𝑦 / 𝑦 ] 𝜑 )
2		sbid	⊢ ( [ 𝑦 / 𝑦 ] 𝜑 ↔ 𝜑 )
3	2	sbbii	⊢ ( [ 𝑦 / 𝑥 ] [ 𝑦 / 𝑦 ] 𝜑 ↔ [ 𝑦 / 𝑥 ] 𝜑 )
4	1 3	bitri	⊢ ( [ 𝑦 / 𝑥 ] [ 𝑥 / 𝑦 ] 𝜑 ↔ [ 𝑦 / 𝑥 ] 𝜑 )