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Theorem sbft 2363
Description: Substitution has no effect on a non-free 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 1870 . . 3 ([𝑦 / 𝑥]𝜑 → ∃𝑥𝜑)
2 19.9t 2057 . . 3 (Ⅎ𝑥𝜑 → (∃𝑥𝜑𝜑))
31, 2syl5ib 232 . 2 (Ⅎ𝑥𝜑 → ([𝑦 / 𝑥]𝜑𝜑))
4 nf5r 2050 . . 3 (Ⅎ𝑥𝜑 → (𝜑 → ∀𝑥𝜑))
5 stdpc4 2337 . . 3 (∀𝑥𝜑 → [𝑦 / 𝑥]𝜑)
64, 5syl6 34 . 2 (Ⅎ𝑥𝜑 → (𝜑 → [𝑦 / 𝑥]𝜑))
73, 6impbid 200 1 (Ⅎ𝑥𝜑 → ([𝑦 / 𝑥]𝜑𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wal 1472  wex 1694  wnf 1698  [wsb 1866
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-12 2032  ax-13 2229
This theorem depends on definitions:  df-bi 195  df-an 384  df-ex 1695  df-nf 1700  df-sb 1867
This theorem is referenced by:  sbf  2364  sbctt  3463  wl-sbrimt  32310  wl-sblimt  32311  wl-equsb4  32317
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