Theorem sb8ef 2387

Description: Substitution of variable in existential quantifier. Version of sb8e 2550 with a disjoint variable condition, not requiring ax-13 2404. (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 2191	. 2 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑
3		sbequ12 2287	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
4	1, 2, 3	cbvexv1 2374	1 ⊢ (∃𝑥𝜑 ↔ ∃𝑦[𝑦 / 𝑥]𝜑)

Syntax hints:   ↔ wb 209  ∃wex 1809  Ⅎwnf 1813  [wsb 2096

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-10 2176  ax-11 2192  ax-12 2213

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-ex 1810  df-nf 1814  df-sb 2097

This theorem is referenced by:  2sb8ef  2388  sbnf2  2390  mo3  2592  bnj985v  35341  regsfromregtco  37069  bj-axseprep  37731  sbcexf  38784