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

Theorem sbft 2268
Description: Substitution has no effect on a nonfree variable. (Contributed by NM, 30-May-2009.) (Revised by Mario Carneiro, 12-Oct-2016.) (Proof shortened by Wolf Lammen, 3-May-2018.)
Assertion
Ref Expression
sbft (Ⅎ𝑥𝜑 → ([𝑦 / 𝑥]𝜑𝜑))

Proof of Theorem sbft
StepHypRef Expression
1 spsbe 2080 . . 3 ([𝑦 / 𝑥]𝜑 → ∃𝑥𝜑)
2 19.9t 2202 . . 3 (Ⅎ𝑥𝜑 → (∃𝑥𝜑𝜑))
31, 2imbitrid 244 . 2 (Ⅎ𝑥𝜑 → ([𝑦 / 𝑥]𝜑𝜑))
4 nf5r 2192 . . 3 (Ⅎ𝑥𝜑 → (𝜑 → ∀𝑥𝜑))
5 stdpc4 2066 . . 3 (∀𝑥𝜑 → [𝑦 / 𝑥]𝜑)
64, 5syl6 35 . 2 (Ⅎ𝑥𝜑 → (𝜑 → [𝑦 / 𝑥]𝜑))
73, 6impbid 212 1 (Ⅎ𝑥𝜑 → ([𝑦 / 𝑥]𝜑𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wal 1535  wex 1776  wnf 1780  [wsb 2062
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-12 2175
This theorem depends on definitions:  df-bi 207  df-ex 1777  df-nf 1781  df-sb 2063
This theorem is referenced by:  sbf  2269  sbctt  3867  wl-sbrimt  37528  wl-sblimt  37529  wl-sb8ft  37531  wl-equsb4  37538  ichnfimlem  47388
  Copyright terms: Public domain W3C validator