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Metamath Proof Explorer

Theorem cbvrexsvw

Description: Change bound variable by using a substitution. Version of cbvrexsv with a disjoint variable condition, which does not require ax-13 . (Contributed by NM, 2-Mar-2008) Avoid ax-13 . (Revised by GG, 10-Jan-2024) (Proof shortened by Wolf Lammen, 8-Mar-2025)

		Ref	Expression
	Assertion	cbvrexsvw	⊢ ( ∃ 𝑥 ∈ 𝐴 𝜑 ↔ ∃ 𝑦 ∈ 𝐴 [ 𝑦 / 𝑥 ] 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		nfv	⊢ Ⅎ 𝑦 𝜑
2		nfs1v	⊢ Ⅎ 𝑥 [ 𝑦 / 𝑥 ] 𝜑
3		sbequ12	⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ [ 𝑦 / 𝑥 ] 𝜑 ) )
4	1 2 3	cbvrexw	⊢ ( ∃ 𝑥 ∈ 𝐴 𝜑 ↔ ∃ 𝑦 ∈ 𝐴 [ 𝑦 / 𝑥 ] 𝜑 )