Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  sb8 GIF version

Theorem sb8 1736
 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 1690 . 2 𝑥[𝑦 / 𝑥]𝜑
3 sbequ12 1654 . 2 (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
41, 2, 3cbval 1637 1 (∀𝑥𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑)
 Colors of variables: wff set class Syntax hints:   ↔ wb 98  ∀wal 1241  Ⅎwnf 1349  [wsb 1645 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427 This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646 This theorem is referenced by:  sbnf2  1857  sb8eu  1913  nfraldya  2358  rabeq0  3247  abeq0  3248  sb8iota  4874
 Copyright terms: Public domain W3C validator