Theorem nfnf1 2160

Description: The setvar 𝑥 is not free in Ⅎ𝑥𝜑. (Contributed by Mario Carneiro, 11-Aug-2016.) Remove dependency on ax-12 2185. (Revised by Wolf Lammen, 12-Oct-2021.)

Assertion

Ref	Expression
nfnf1	⊢ Ⅎ𝑥Ⅎ𝑥𝜑

Proof of Theorem nfnf1

Step	Hyp	Ref	Expression
1		df-nf 1786	. 2 ⊢ (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
2		nfe1 2156	. . 3 ⊢ Ⅎ𝑥∃𝑥𝜑
3		nfa1 2157	. . 3 ⊢ Ⅎ𝑥∀𝑥𝜑
4	2, 3	nfim 1898	. 2 ⊢ Ⅎ𝑥(∃𝑥𝜑 → ∀𝑥𝜑)
5	1, 4	nfxfr 1855	1 ⊢ Ⅎ𝑥Ⅎ𝑥𝜑

Syntax hints:   → wi 4  ∀wal 1540  ∃wex 1781  Ⅎwnf 1785

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-10 2147

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-ex 1782  df-nf 1786

This theorem is referenced by:  nfsb4t  2504  nfnfc1  2902  sbcnestgfw  4375  sbcnestgf  4380  bj-sbf4  37115  wl-equsal1t  37826  wl-sbid2ft  37829  wl-sb8t  37836  wl-mo2tf  37855  wl-eutf  37857  wl-mo2t  37859  wl-mo3t  37860  wl-sb8eut  37862  wl-sb8eutv  37863  ichnfimlem  47852  ichnfim  47853