Theorem nfnf1 2153

Description: The setvar 𝑥 is not free in Ⅎ𝑥𝜑. (Contributed by Mario Carneiro, 11-Aug-2016.) Remove dependency on ax-12 2173. (Revised by Wolf Lammen, 12-Oct-2021.)

Assertion

Ref	Expression
nfnf1	⊢ Ⅎ𝑥Ⅎ𝑥𝜑

Proof of Theorem nfnf1

Step	Hyp	Ref	Expression
1		df-nf 1788	. 2 ⊢ (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
2		nfe1 2149	. . 3 ⊢ Ⅎ𝑥∃𝑥𝜑
3		nfa1 2150	. . 3 ⊢ Ⅎ𝑥∀𝑥𝜑
4	2, 3	nfim 1900	. 2 ⊢ Ⅎ𝑥(∃𝑥𝜑 → ∀𝑥𝜑)
5	1, 4	nfxfr 1856	1 ⊢ Ⅎ𝑥Ⅎ𝑥𝜑

Syntax hints:   → wi 4  ∀wal 1537  ∃wex 1783  Ⅎwnf 1787

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-10 2139

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-ex 1784  df-nf 1788

This theorem is referenced by:  sb4bOLD  2476  nfsb4t  2503  nfnfc1  2909  nfabdwOLD  2930  sbcnestgfw  4349  sbcnestgf  4354  bj-sbf4  34950  wl-equsal1t  35627  wl-sb6rft  35630  wl-sb8t  35634  wl-mo2tf  35653  wl-eutf  35655  wl-mo2t  35657  wl-mo3t  35658  wl-sb8eut  35659  ichnfimlem  44803  ichnfim  44804