Theorem sb8ev 2322

Description: Substitution of variable in existential quantifier. Version of sb8e 2502 with a disjoint variable condition, not requiring ax-13 2334. (Contributed by Wolf Lammen, 19-Jan-2023.)

Hypothesis

Ref	Expression
sb8v.nf	⊢ Ⅎ𝑦𝜑

Assertion

Ref	Expression
sb8ev	⊢ (∃𝑥𝜑 ↔ ∃𝑦[𝑦 / 𝑥]𝜑)

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem sb8ev

Step	Hyp	Ref	Expression
1		sb8v.nf	. 2 ⊢ Ⅎ𝑦𝜑
2		nfs1v 2254	. 2 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑
3		sbequ12 2229	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
4	1, 2, 3	cbvexv1 2313	1 ⊢ (∃𝑥𝜑 ↔ ∃𝑦[𝑦 / 𝑥]𝜑)

Syntax hints:   ↔ wb 198  ∃wex 1823  Ⅎwnf 1827  [wsb 2011

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-10 2135  ax-11 2150  ax-12 2163

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-ex 1824  df-nf 1828  df-sb 2012

This theorem is referenced by:  2sb8ev  2323  2sb8evOLD  2324  sbnf2  2326  mo3  2580