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Theorem nfr 1452
Description: Consequence of the definition of not-free. (Contributed by Mario Carneiro, 26-Sep-2016.)
Assertion
Ref Expression
nfr (Ⅎ𝑥𝜑 → (𝜑 → ∀𝑥𝜑))

Proof of Theorem nfr
StepHypRef Expression
1 df-nf 1391 . 2 (Ⅎ𝑥𝜑 ↔ ∀𝑥(𝜑 → ∀𝑥𝜑))
2 sp 1442 . 2 (∀𝑥(𝜑 → ∀𝑥𝜑) → (𝜑 → ∀𝑥𝜑))
31, 2sylbi 119 1 (Ⅎ𝑥𝜑 → (𝜑 → ∀𝑥𝜑))
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1283  wnf 1390
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-4 1441
This theorem depends on definitions:  df-bi 115  df-nf 1391
This theorem is referenced by:  nfri  1453  nfrd  1454  nfimd  1518  19.23t  1608  equs5or  1752  sbequi  1761  sbft  1770  sbcomxyyz  1888  rgen2a  2418
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