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Theorem nfnf1 2071
 Description: The setvar 𝑥 is not free in Ⅎ𝑥𝜑. (Contributed by Mario Carneiro, 11-Aug-2016.) Remove dependency on ax-12 2087. (Revised by Wolf Lammen, 12-Oct-2021.)
Assertion
Ref Expression
nfnf1 𝑥𝑥𝜑

Proof of Theorem nfnf1
StepHypRef Expression
1 df-nf 1750 . 2 (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
2 nfe1 2067 . . 3 𝑥𝑥𝜑
3 nfa1 2068 . . 3 𝑥𝑥𝜑
42, 3nfim 1865 . 2 𝑥(∃𝑥𝜑 → ∀𝑥𝜑)
51, 4nfxfr 1819 1 𝑥𝑥𝜑
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1521  ∃wex 1744  Ⅎwnf 1748 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-10 2059 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-ex 1745  df-nf 1750 This theorem is referenced by:  nfaldOLD  2202  nfeqf2  2333  nfsb4t  2417  nfnfc1  2796  sbcnestgf  4028  dfnfc2OLD  4487  bj-sbf4  32952  wl-equsal1t  33457  wl-sb6rft  33460  wl-sb8t  33463  wl-mo2tf  33483  wl-eutf  33485  wl-mo2t  33487  wl-mo3t  33488  wl-sb8eut  33489
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