Theorem nfnf1 2154

Description: The setvar 𝑥 is not free in Ⅎ𝑥𝜑. (Contributed by Mario Carneiro, 11-Aug-2016.) Remove dependency on ax-12 2174. (Revised by Wolf Lammen, 12-Oct-2021.)

Assertion

Ref	Expression
nfnf1	⊢ Ⅎ𝑥Ⅎ𝑥𝜑

Proof of Theorem nfnf1

Step	Hyp	Ref	Expression
1		df-nf 1790	. 2 ⊢ (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
2		nfe1 2150	. . 3 ⊢ Ⅎ𝑥∃𝑥𝜑
3		nfa1 2151	. . 3 ⊢ Ⅎ𝑥∀𝑥𝜑
4	2, 3	nfim 1902	. 2 ⊢ Ⅎ𝑥(∃𝑥𝜑 → ∀𝑥𝜑)
5	1, 4	nfxfr 1858	1 ⊢ Ⅎ𝑥Ⅎ𝑥𝜑

Syntax hints:   → wi 4  ∀wal 1539  ∃wex 1785  Ⅎwnf 1789

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-10 2140

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-ex 1786  df-nf 1790

This theorem is referenced by:  sb4bOLD  2477  nfsb4t  2504  nfnfc1  2911  nfabdwOLD  2932  sbcnestgfw  4357  sbcnestgf  4362  bj-sbf4  35002  wl-equsal1t  35679  wl-sb6rft  35682  wl-sb8t  35686  wl-mo2tf  35705  wl-eutf  35707  wl-mo2t  35709  wl-mo3t  35710  wl-sb8eut  35711  ichnfimlem  44867  ichnfim  44868