Theorem sb8ev 2353

Description: Substitution of variable in existential quantifier. Version of sb8e 2522 with a disjoint variable condition, not requiring ax-13 2372. (Contributed by NM, 12-Aug-1993.) (Revised 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 2155	. 2 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑
3		sbequ12 2247	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
4	1, 2, 3	cbvexv1 2341	1 ⊢ (∃𝑥𝜑 ↔ ∃𝑦[𝑦 / 𝑥]𝜑)

Syntax hints:   ↔ wb 205  ∃wex 1783  Ⅎwnf 1787  [wsb 2068

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-10 2139  ax-11 2156  ax-12 2173

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-ex 1784  df-nf 1788  df-sb 2069

This theorem is referenced by:  2sb8ev  2354  sbnf2  2356  mo3  2564  cbvmowOLD  2604  bnj985v  32833  sbcexf  36200