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Theorem nf5r 2052
 Description: Consequence of the definition of not-free. (Contributed by Mario Carneiro, 26-Sep-2016.) df-nf 1701 changed. (Revised by Wolf Lammen, 11-Sep-2021.)
Assertion
Ref Expression
nf5r (Ⅎ𝑥𝜑 → (𝜑 → ∀𝑥𝜑))

Proof of Theorem nf5r
StepHypRef Expression
1 19.8a 2039 . 2 (𝜑 → ∃𝑥𝜑)
2 df-nf 1701 . . 3 (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
32biimpi 205 . 2 (Ⅎ𝑥𝜑 → (∃𝑥𝜑 → ∀𝑥𝜑))
41, 3syl5 33 1 (Ⅎ𝑥𝜑 → (𝜑 → ∀𝑥𝜑))
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1473  ∃wex 1695  Ⅎwnf 1699 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-12 2034 This theorem depends on definitions:  df-bi 196  df-ex 1696  df-nf 1701 This theorem is referenced by:  nf5ri  2053  nf5rd  2054  19.21t  2061  sbft  2367  bj-alrim  31870  bj-nexdt  31874  bj-cbv3tb  31898  bj-nfs1t2  31902  bj-sbftv  31951  bj-equsal1t  31997  stdpc5t  32002  bj-axc14  32032  wl-nfeqfb  32502
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