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Theorem sbn1 39210
Description: One direction of sbn 2287, using fewer axioms. Compare 19.2 1981. (Contributed by Steven Nguyen, 18-Aug-2023.)
Assertion
Ref Expression
sbn1 ([𝑡 / 𝑥] ¬ 𝜑 → ¬ [𝑡 / 𝑥]𝜑)

Proof of Theorem sbn1
StepHypRef Expression
1 fal 1551 . . 3 ¬ ⊥
21nsb 39209 . 2 ¬ [𝑡 / 𝑥]⊥
3 pm2.21 123 . . 3 𝜑 → (𝜑 → ⊥))
43sb2imi 2080 . 2 ([𝑡 / 𝑥] ¬ 𝜑 → ([𝑡 / 𝑥]𝜑 → [𝑡 / 𝑥]⊥))
52, 4mtoi 201 1 ([𝑡 / 𝑥] ¬ 𝜑 → ¬ [𝑡 / 𝑥]𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wfal 1549  [wsb 2069
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970
This theorem depends on definitions:  df-bi 209  df-tru 1540  df-fal 1550  df-ex 1781  df-sb 2070
This theorem is referenced by: (None)
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