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Theorem nfnf1 2158
 Description: The setvar 𝑥 is not free in Ⅎ𝑥𝜑. (Contributed by Mario Carneiro, 11-Aug-2016.) Remove dependency on ax-12 2178. (Revised by Wolf Lammen, 12-Oct-2021.)
Assertion
Ref Expression
nfnf1 𝑥𝑥𝜑

Proof of Theorem nfnf1
StepHypRef Expression
1 df-nf 1786 . 2 (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
2 nfe1 2154 . . 3 𝑥𝑥𝜑
3 nfa1 2155 . . 3 𝑥𝑥𝜑
42, 3nfim 1897 . 2 𝑥(∃𝑥𝜑 → ∀𝑥𝜑)
51, 4nfxfr 1854 1 𝑥𝑥𝜑
 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-10 2145 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-nf 1786 This theorem is referenced by:  sb4bOLD  2501  nfsb4t  2539  nfsb4tALT  2604  nfnfc1  2982  nfabdw  3000  sbcnestgfw  4342  sbcnestgf  4347  bj-sbf4  34239  wl-equsal1t  34905  wl-sb6rft  34908  wl-sb8t  34912  wl-mo2tf  34931  wl-eutf  34933  wl-mo2t  34935  wl-mo3t  34936  wl-sb8eut  34937  ichnfimlem  43920  ichnfim  43921
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