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Metamath Proof Explorer

Theorem sb8ef

Description: Substitution of variable in existential quantifier. Version of sb8e with a disjoint variable condition, not requiring ax-13 . (Contributed by NM, 12-Aug-1993) (Revised by Wolf Lammen, 19-Jan-2023)

		Ref	Expression
	Hypothesis	sb8f.nf	⊢ Ⅎ 𝑦 𝜑
	Assertion	sb8ef	⊢ ( ∃ 𝑥 𝜑 ↔ ∃ 𝑦 [ 𝑦 / 𝑥 ] 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		sb8f.nf	⊢ Ⅎ 𝑦 𝜑
2		nfs1v	⊢ Ⅎ 𝑥 [ 𝑦 / 𝑥 ] 𝜑
3		sbequ12	⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ [ 𝑦 / 𝑥 ] 𝜑 ) )
4	1 2 3	cbvexv1	⊢ ( ∃ 𝑥 𝜑 ↔ ∃ 𝑦 [ 𝑦 / 𝑥 ] 𝜑 )