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Theorem sb8 1778
Description: Substitution of variable in universal quantifier. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Jim Kingdon, 15-Jan-2018.)
Hypothesis
Ref Expression
sb8e.1 𝑦𝜑
Assertion
Ref Expression
sb8 (∀𝑥𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑)

Proof of Theorem sb8
StepHypRef Expression
1 sb8e.1 . 2 𝑦𝜑
21nfs1 1731 . 2 𝑥[𝑦 / 𝑥]𝜑
3 sbequ12 1695 . 2 (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
41, 2, 3cbval 1678 1 (∀𝑥𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑)
Colors of variables: wff set class
Syntax hints:  wb 103  wal 1283  wnf 1390  [wsb 1686
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-11 1438  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1687
This theorem is referenced by:  sbnf2  1899  sb8eu  1955  nfraldya  2401  rabeq0  3281  abeq0  3282  sb8iota  4904
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