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Theorem bj-sbft 34220
Description: Version of sbft 2269 using Ⅎ', proved from core axioms. (Contributed by BJ, 19-Nov-2023.)
Assertion
Ref Expression
bj-sbft (Ⅎ'𝑥𝜑 → ([𝑡 / 𝑥]𝜑𝜑))

Proof of Theorem bj-sbft
StepHypRef Expression
1 spsbe 2087 . . 3 ([𝑡 / 𝑥]𝜑 → ∃𝑥𝜑)
2 bj-nnfe 34178 . . 3 (Ⅎ'𝑥𝜑 → (∃𝑥𝜑𝜑))
31, 2syl5 34 . 2 (Ⅎ'𝑥𝜑 → ([𝑡 / 𝑥]𝜑𝜑))
4 bj-nnfa 34176 . . 3 (Ⅎ'𝑥𝜑 → (𝜑 → ∀𝑥𝜑))
5 stdpc4 2073 . . 3 (∀𝑥𝜑 → [𝑡 / 𝑥]𝜑)
64, 5syl6 35 . 2 (Ⅎ'𝑥𝜑 → (𝜑 → [𝑡 / 𝑥]𝜑))
73, 6impbid 215 1 (Ⅎ'𝑥𝜑 → ([𝑡 / 𝑥]𝜑𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wal 1536  wex 1781  [wsb 2069  Ⅎ'wnnf 34171
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070  df-bj-nnf 34172
This theorem is referenced by: (None)
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