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Theorem sb8ev 2375
Description: Substitution of variable in existential quantifier. Version of sb8e 2560 with a disjoint variable condition, not requiring ax-13 2391. (Contributed by NM, 12-Aug-1993.) (Revised by Wolf Lammen, 19-Jan-2023.)
Hypothesis
Ref Expression
sb8v.nf 𝑦𝜑
Assertion
Ref Expression
sb8ev (∃𝑥𝜑 ↔ ∃𝑦[𝑦 / 𝑥]𝜑)
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem sb8ev
StepHypRef Expression
1 sb8v.nf . 2 𝑦𝜑
2 nfs1v 2160 . 2 𝑥[𝑦 / 𝑥]𝜑
3 sbequ12 2254 . 2 (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
41, 2, 3cbvexv1 2363 1 (∃𝑥𝜑 ↔ ∃𝑦[𝑦 / 𝑥]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 209  wex 1781  wnf 1785  [wsb 2069
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-10 2145  ax-11 2161  ax-12 2178
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-nf 1786  df-sb 2070
This theorem is referenced by:  2sb8ev  2376  sbnf2  2378  mo3  2647  cbvmowOLD  2688  bnj985v  32299  sbcexf  35515
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