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Theorem nfnf1 2143
Description: The setvar 𝑥 is not free in 𝑥𝜑. (Contributed by Mario Carneiro, 11-Aug-2016.) Remove dependency on ax-12 2166. (Revised by Wolf Lammen, 12-Oct-2021.)
Assertion
Ref Expression
nfnf1 𝑥𝑥𝜑

Proof of Theorem nfnf1
StepHypRef Expression
1 df-nf 1778 . 2 (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
2 nfe1 2139 . . 3 𝑥𝑥𝜑
3 nfa1 2140 . . 3 𝑥𝑥𝜑
42, 3nfim 1891 . 2 𝑥(∃𝑥𝜑 → ∀𝑥𝜑)
51, 4nfxfr 1847 1 𝑥𝑥𝜑
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-10 2129
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-ex 1774  df-nf 1778
This theorem is referenced by:  nfsb4t  2492  nfnfc1  2894  nfabdwOLD  2916  sbcnestgfw  4420  sbcnestgf  4425  bj-sbf4  36448  wl-equsal1t  37140  wl-sbid2ft  37143  wl-sb8t  37150  wl-mo2tf  37169  wl-eutf  37171  wl-mo2t  37173  wl-mo3t  37174  wl-sb8eut  37176  wl-sb8eutv  37177  ichnfimlem  46940  ichnfim  46941
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