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Theorem sb8 2514
Description: Substitution of variable in universal quantifier. (Contributed by NM, 16-May-1993.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Jim Kingdon, 15-Jan-2018.)
Hypothesis
Ref Expression
sb5rf.1 𝑦𝜑
Assertion
Ref Expression
sb8 (∀𝑥𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑)

Proof of Theorem sb8
StepHypRef Expression
1 sb5rf.1 . 2 𝑦𝜑
21nfs1 2453 . 2 𝑥[𝑦 / 𝑥]𝜑
3 sbequ12 2276 . 2 (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
41, 2, 3cbval 2373 1 (∀𝑥𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 197  wal 1650  wnf 1878  [wsb 2061
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2349
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-ex 1875  df-nf 1879  df-sb 2062
This theorem is referenced by:  sbhb  2531  sbnf2  2532  sb8eu  2623  abv  3358  sb8iota  6037  mo5f  29714  ax11-pm2  33184  bj-nfcf  33279  wl-sb8eut  33716  sbcalf  34271
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