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Theorem nfr 1511
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 1454 . 2 (Ⅎ𝑥𝜑 ↔ ∀𝑥(𝜑 → ∀𝑥𝜑))
2 sp 1504 . 2 (∀𝑥(𝜑 → ∀𝑥𝜑) → (𝜑 → ∀𝑥𝜑))
31, 2sylbi 120 1 (Ⅎ𝑥𝜑 → (𝜑 → ∀𝑥𝜑))
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1346  wnf 1453
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-4 1503
This theorem depends on definitions:  df-bi 116  df-nf 1454
This theorem is referenced by:  nfri  1512  nfrd  1513  nfimd  1578  19.23t  1670  equs5or  1823  sbequi  1832  sbft  1841  sbcomxyyz  1965  rgen2a  2524
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