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Theorem sbft 2253
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 2077 . . 3 ([𝑦 / 𝑥]𝜑 → ∃𝑥𝜑)
2 19.9t 2189 . . 3 (Ⅎ𝑥𝜑 → (∃𝑥𝜑𝜑))
31, 2imbitrid 243 . 2 (Ⅎ𝑥𝜑 → ([𝑦 / 𝑥]𝜑𝜑))
4 nf5r 2179 . . 3 (Ⅎ𝑥𝜑 → (𝜑 → ∀𝑥𝜑))
5 stdpc4 2063 . . 3 (∀𝑥𝜑 → [𝑦 / 𝑥]𝜑)
64, 5syl6 35 . 2 (Ⅎ𝑥𝜑 → (𝜑 → [𝑦 / 𝑥]𝜑))
73, 6impbid 211 1 (Ⅎ𝑥𝜑 → ([𝑦 / 𝑥]𝜑𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wal 1531  wex 1773  wnf 1777  [wsb 2059
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-12 2163
This theorem depends on definitions:  df-bi 206  df-ex 1774  df-nf 1778  df-sb 2060
This theorem is referenced by:  sbf  2254  sbctt  3846  wl-sbrimt  36906  wl-sblimt  36907  wl-sb8ft  36909  wl-equsb4  36916  ichnfimlem  46641
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