MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  sb8 Structured version   Visualization version   GIF version

Theorem sb8 2452
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 2393 . 2 𝑥[𝑦 / 𝑥]𝜑
3 sbequ12 2149 . 2 (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
41, 2, 3cbval 2307 1 (∀𝑥𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 196  wal 1521  wnf 1748  [wsb 1937
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-ex 1745  df-nf 1750  df-sb 1938
This theorem is referenced by:  sbhb  2466  sbnf2  2467  sb8eu  2532  abv  3237  sb8iota  5896  mo5f  29452  ax11-pm2  32948  bj-nfcf  33045  wl-sb8eut  33489  sbcalf  34047
  Copyright terms: Public domain W3C validator