Theorem nfnf1 2165

Description: The setvar 𝑥 is not free in Ⅎ𝑥𝜑. (Contributed by Mario Carneiro, 11-Aug-2016.) Remove dependency on ax-12 2189. (Revised by Wolf Lammen, 12-Oct-2021.)

Assertion

Ref	Expression
nfnf1	⊢ Ⅎ𝑥Ⅎ𝑥𝜑

Proof of Theorem nfnf1

Step	Hyp	Ref	Expression
1		df-nf 1791	. 2 ⊢ (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
2		nfe1 2161	. . 3 ⊢ Ⅎ𝑥∃𝑥𝜑
3		nfa1 2162	. . 3 ⊢ Ⅎ𝑥∀𝑥𝜑
4	2, 3	nfim 1903	. 2 ⊢ Ⅎ𝑥(∃𝑥𝜑 → ∀𝑥𝜑)
5	1, 4	nfxfr 1860	1 ⊢ Ⅎ𝑥Ⅎ𝑥𝜑

Syntax hints:   → wi 4  ∀wal 1545  ∃wex 1786  Ⅎwnf 1790

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-10 2152

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-ex 1787  df-nf 1791

This theorem is referenced by:  nfsb4t  2507  nfnfc1  2904  sbcnestgfw  4350  sbcnestgf  4355  bj-sbf4  37202  wl-equsal1t  37922  wl-sbid2ft  37925  wl-sb8t  37932  wl-mo2tf  37951  wl-eutf  37953  wl-mo2t  37955  wl-mo3t  37956  wl-sb8eut  37958  wl-sb8eutv  37959  ichnfimlem  47946  ichnfim  47947