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Theorem nsb 2112
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 1788 . . 3 (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑)
21biimpi 219 . 2 (∀𝑥 ¬ 𝜑 → ¬ ∃𝑥𝜑)
3 spsbe 2092 . 2 ([𝑡 / 𝑥]𝜑 → ∃𝑥𝜑)
42, 3nsyl 142 1 (∀𝑥 ¬ 𝜑 → ¬ [𝑡 / 𝑥]𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1540  wex 1786  [wsb 2074
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975
This theorem depends on definitions:  df-bi 210  df-ex 1787  df-sb 2075
This theorem is referenced by:  sbn1  2113  noel  4219  sbn1ALT  34673
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