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Theorem nf5r 2179
Description: Consequence of the definition of not-free. (Contributed by Mario Carneiro, 26-Sep-2016.) df-nf 1778 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 2166 . 2 (𝜑 → ∃𝑥𝜑)
2 id 22 . . 3 (Ⅎ𝑥𝜑 → Ⅎ𝑥𝜑)
32nfrd 1785 . 2 (Ⅎ𝑥𝜑 → (∃𝑥𝜑 → ∀𝑥𝜑))
41, 3syl5 34 1 (Ⅎ𝑥𝜑 → (𝜑 → ∀𝑥𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1531  wex 1773  wnf 1777
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-12 2163
This theorem depends on definitions:  df-bi 206  df-ex 1774  df-nf 1778
This theorem is referenced by:  nf5rd  2181  19.3t  2186  sbft  2253  bj-alrim  36062  bj-nexdt  36066  bj-cbv3tb  36156  bj-nfs1t2  36160  bj-equsal1t  36191  stdpc5t  36196  bj-axc14  36226  wl-nfeqfb  36896
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