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Theorem nfnf1 2196
Description: The setvar 𝑥 is not free in 𝑥𝜑. (Contributed by Mario Carneiro, 11-Aug-2016.) Remove dependency on ax-12 2211. (Revised by Wolf Lammen, 12-Oct-2021.)
Assertion
Ref Expression
nfnf1 𝑥𝑥𝜑

Proof of Theorem nfnf1
StepHypRef Expression
1 df-nf 1879 . 2 (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
2 nfe1 2192 . . 3 𝑥𝑥𝜑
3 nfa1 2193 . . 3 𝑥𝑥𝜑
42, 3nfim 1995 . 2 𝑥(∃𝑥𝜑 → ∀𝑥𝜑)
51, 4nfxfr 1948 1 𝑥𝑥𝜑
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1650  wex 1874  wnf 1878
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-10 2183
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-ex 1875  df-nf 1879
This theorem is referenced by:  nfeqf2OLD  2397  nfsb4t  2480  nfnfc1  2910  sbcnestgf  4158  bj-sbf4  33259  wl-equsal1t  33755  wl-sb6rft  33759  wl-sb8t  33762  wl-mo2tf  33781  wl-eutf  33783  wl-mo2t  33785  wl-mo3t  33786  wl-sb8eut  33787
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