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Theorem sb8 2556
Description: Substitution of variable in universal quantifier. For a version requiring disjoint variables, but fewer axioms, see sb8v 2376. (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 2496 . 2 𝑥[𝑦 / 𝑥]𝜑
3 sbequ12 2286 . 2 (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
41, 2, 3cbval 2423 1 (∀𝑥𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 198  wal 1654  wnf 1882  [wsb 2067
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-ex 1879  df-nf 1883  df-sb 2068
This theorem is referenced by:  sbhb  2573  sbnf2OLD  2574  abv  3423  sb8iota  6093  mo5f  29868  ax11-pm2  33340  bj-nfcf  33436  wl-sb8eut  33896  sbcalf  34451
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