Theorem nfnf1 2143

Description: The setvar 𝑥 is not free in Ⅎ𝑥𝜑. (Contributed by Mario Carneiro, 11-Aug-2016.) Remove dependency on ax-12 2166. (Revised by Wolf Lammen, 12-Oct-2021.)

Assertion

Ref	Expression
nfnf1	⊢ Ⅎ𝑥Ⅎ𝑥𝜑

Proof of Theorem nfnf1

Step	Hyp	Ref	Expression
1		df-nf 1778	. 2 ⊢ (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
2		nfe1 2139	. . 3 ⊢ Ⅎ𝑥∃𝑥𝜑
3		nfa1 2140	. . 3 ⊢ Ⅎ𝑥∀𝑥𝜑
4	2, 3	nfim 1891	. 2 ⊢ Ⅎ𝑥(∃𝑥𝜑 → ∀𝑥𝜑)
5	1, 4	nfxfr 1847	1 ⊢ Ⅎ𝑥Ⅎ𝑥𝜑

Syntax hints:   → wi 4  ∀wal 1531  ∃wex 1773  Ⅎwnf 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