Theorem sb8ef 2389

Description: Substitution of variable in existential quantifier. Version of sb8e 2552 with a disjoint variable condition, not requiring ax-13 2406. (Contributed by NM, 12-Aug-1993.) (Revised by Wolf Lammen, 19-Jan-2023.)

Hypothesis

Ref	Expression
sb8f.nf	⊢ Ⅎ𝑦𝜑

Assertion

Ref	Expression
sb8ef	⊢ (∃𝑥𝜑 ↔ ∃𝑦[𝑦 / 𝑥]𝜑)

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥, 𝑦)

Proof of Theorem sb8ef

Step	Hyp	Ref	Expression
1		sb8f.nf	. 2 ⊢ Ⅎ𝑦𝜑
2		nfs1v 2194	. 2 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑
3		sbequ12 2289	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
4	1, 2, 3	cbvexv1 2376	1 ⊢ (∃𝑥𝜑 ↔ ∃𝑦[𝑦 / 𝑥]𝜑)

This proof depends on syntax axioms:   ↔ wb 209  ∃wex 1812  Ⅎwnf 1816  [wsb 2099

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-10 2179  ax-11 2195  ax-12 2216

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ex 1813  df-nf 1817  df-sb 2100

This theorem is used by:  2sb8ef  2390  sbnf2  2392  mo3  2594  bnj985v  35408  regsfromregtco  37108  bj-axseprep  37770  sbcexf  38824