Theorem sb8e

Description: Substitution of variable in existential quantifier. Usage of this theorem is discouraged because it depends on ax-13 . For a version requiring disjoint variables, but fewer axioms, see sb8ev . (Contributed by NM, 12-Aug-1993) (Revised by Mario Carneiro, 6-Oct-2016) (Proof shortened by Jim Kingdon, 15-Jan-2018) (New usage is discouraged.)

	Ref	Expression
Hypothesis	sb8.1	⊢ Ⅎ 𝑦 𝜑
Assertion	sb8e	⊢ ( ∃ 𝑥 𝜑 ↔ ∃ 𝑦 [ 𝑦 / 𝑥 ] 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		sb8.1	⊢ Ⅎ 𝑦 𝜑
2	1	nfs1	⊢ Ⅎ 𝑥 [ 𝑦 / 𝑥 ] 𝜑
3		sbequ12	⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ [ 𝑦 / 𝑥 ] 𝜑 ) )
4	1 2 3	cbvex	⊢ ( ∃ 𝑥 𝜑 ↔ ∃ 𝑦 [ 𝑦 / 𝑥 ] 𝜑 )