Theorem sb8ev 2374

Description: Substitution of variable in existential quantifier. Version of sb8e 2560 with a disjoint variable condition, not requiring ax-13 2390. (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 2160	. 2 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑
3		sbequ12 2253	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
4	1, 2, 3	cbvexv1 2362	1 ⊢ (∃𝑥𝜑 ↔ ∃𝑦[𝑦 / 𝑥]𝜑)

Syntax hints:   ↔ wb 208  ∃wex 1780  Ⅎwnf 1784  [wsb 2069

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-10 2145  ax-11 2161  ax-12 2177

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ex 1781  df-nf 1785  df-sb 2070

This theorem is referenced by:  2sb8ev  2375  sbnf2  2377  mo3  2648  cbvmow  2688  bnj985v  32225  sbcexf  35408