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Theorem nf5r 2194
 Description: Consequence of the definition of not-free. (Contributed by Mario Carneiro, 26-Sep-2016.) df-nf 1786 changed. (Revised by Wolf Lammen, 11-Sep-2021.) (Proof shortened by Wolf Lammen, 23-Nov-2023.)
Assertion
Ref Expression
nf5r (Ⅎ𝑥𝜑 → (𝜑 → ∀𝑥𝜑))

Proof of Theorem nf5r
StepHypRef Expression
1 19.8a 2181 . 2 (𝜑 → ∃𝑥𝜑)
2 id 22 . . 3 (Ⅎ𝑥𝜑 → Ⅎ𝑥𝜑)
32nfrd 1793 . 2 (Ⅎ𝑥𝜑 → (∃𝑥𝜑 → ∀𝑥𝜑))
41, 3syl5 34 1 (Ⅎ𝑥𝜑 → (𝜑 → ∀𝑥𝜑))
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1536  ∃wex 1781  Ⅎwnf 1785 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-12 2178 This theorem depends on definitions:  df-bi 210  df-ex 1782  df-nf 1786 This theorem is referenced by:  nf5rd  2197  19.3t  2202  sbft  2271  sbftALT  2593  bj-alrim  34101  bj-nexdt  34105  bj-cbv3tb  34185  bj-nfs1t2  34189  bj-equsal1t  34221  stdpc5t  34226  bj-axc14  34256  wl-nfeqfb  34899
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