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Theorem sb8ef 2347
Description: Substitution of variable in existential quantifier. Version of sb8e 2513 with a disjoint variable condition, not requiring ax-13 2367. (Contributed by NM, 12-Aug-1993.) (Revised by Wolf Lammen, 19-Jan-2023.)
Hypothesis
Ref Expression
sb8f.nf 𝑦𝜑
Assertion
Ref Expression
sb8ef (∃𝑥𝜑 ↔ ∃𝑦[𝑦 / 𝑥]𝜑)
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem sb8ef
StepHypRef Expression
1 sb8f.nf . 2 𝑦𝜑
2 nfs1v 2146 . 2 𝑥[𝑦 / 𝑥]𝜑
3 sbequ12 2239 . 2 (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
41, 2, 3cbvexv1 2334 1 (∃𝑥𝜑 ↔ ∃𝑦[𝑦 / 𝑥]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 205  wex 1774  wnf 1778  [wsb 2060
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-10 2130  ax-11 2147  ax-12 2167
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-ex 1775  df-nf 1779  df-sb 2061
This theorem is referenced by:  2sb8ef  2348  sbnf2  2350  mo3  2554  cbvmowOLD  2594  bnj985v  34584  sbcexf  37588
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