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Theorem sbft 1804
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 1798 . . 3 ([𝑦 / 𝑥]𝜑 → ∃𝑥𝜑)
2 19.9t 1606 . . 3 (Ⅎ𝑥𝜑 → (∃𝑥𝜑𝜑))
31, 2syl5ib 153 . 2 (Ⅎ𝑥𝜑 → ([𝑦 / 𝑥]𝜑𝜑))
4 nfr 1483 . . 3 (Ⅎ𝑥𝜑 → (𝜑 → ∀𝑥𝜑))
5 stdpc4 1733 . . 3 (∀𝑥𝜑 → [𝑦 / 𝑥]𝜑)
64, 5syl6 33 . 2 (Ⅎ𝑥𝜑 → (𝜑 → [𝑦 / 𝑥]𝜑))
73, 6impbid 128 1 (Ⅎ𝑥𝜑 → ([𝑦 / 𝑥]𝜑𝜑))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104  wal 1314  wnf 1421  wex 1453  [wsb 1720
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1408  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-4 1472  ax-i9 1495  ax-ial 1499
This theorem depends on definitions:  df-bi 116  df-nf 1422  df-sb 1721
This theorem is referenced by:  sbctt  2947
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