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Theorem sb8v 2374
 Description: Substitution of variable in universal quantifier. Version of sb8 2560 with a disjoint variable condition, not requiring ax-13 2391. (Contributed by NM, 16-May-1993.) (Revised by Wolf Lammen, 19-Jan-2023.)
Hypothesis
Ref Expression
sb8v.nf 𝑦𝜑
Assertion
Ref Expression
sb8v (∀𝑥𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑)
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem sb8v
StepHypRef Expression
1 sb8v.nf . 2 𝑦𝜑
2 nfs1v 2161 . 2 𝑥[𝑦 / 𝑥]𝜑
3 sbequ12 2254 . 2 (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
41, 2, 3cbvalv1 2362 1 (∀𝑥𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209  ∀wal 1536  Ⅎwnf 1785  [wsb 2070 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 1912  ax-6 1971  ax-7 2016  ax-10 2146  ax-11 2162  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 2071 This theorem is referenced by:  sbnf2  2378  sb8eulem  2684  abv  3481  mo5f  30237  ax11-pm2  34166  bj-nfcf  34258  sbcalf  35430
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