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Metamath Proof Explorer

Theorem sbco2vv

Description: A composition law for substitution. Version of sbco2 with disjoint variable conditions and fewer axioms. (Contributed by NM, 30-Jun-1994) (Revised by BJ, 22-Dec-2020) (Proof shortened by Wolf Lammen, 29-Apr-2023)

		Ref	Expression
	Assertion	sbco2vv	⊢ ( [ 𝑦 / 𝑧 ] [ 𝑧 / 𝑥 ] 𝜑 ↔ [ 𝑦 / 𝑥 ] 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		sbequ	⊢ ( 𝑧 = 𝑤 → ( [ 𝑧 / 𝑥 ] 𝜑 ↔ [ 𝑤 / 𝑥 ] 𝜑 ) )
2		sbequ	⊢ ( 𝑤 = 𝑦 → ( [ 𝑤 / 𝑥 ] 𝜑 ↔ [ 𝑦 / 𝑥 ] 𝜑 ) )
3	1 2	sbievw2	⊢ ( [ 𝑦 / 𝑧 ] [ 𝑧 / 𝑥 ] 𝜑 ↔ [ 𝑦 / 𝑥 ] 𝜑 )