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Theorem nsb 2104
Description: Any substitution in an always false formula is false. (Contributed by Steven Nguyen, 3-May-2023.)
Assertion
Ref Expression
nsb (∀𝑥 ¬ 𝜑 → ¬ [𝑡 / 𝑥]𝜑)

Proof of Theorem nsb
StepHypRef Expression
1 alnex 1784 . . 3 (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑)
21biimpi 215 . 2 (∀𝑥 ¬ 𝜑 → ¬ ∃𝑥𝜑)
3 spsbe 2085 . 2 ([𝑡 / 𝑥]𝜑 → ∃𝑥𝜑)
42, 3nsyl 140 1 (∀𝑥 ¬ 𝜑 → ¬ [𝑡 / 𝑥]𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1537  wex 1782  [wsb 2067
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971
This theorem depends on definitions:  df-bi 206  df-ex 1783  df-sb 2068
This theorem is referenced by:  sbn1  2105  noel  4264  sbn1ALT  35042
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