Theorem nfnf1 2151

Description: The setvar 𝑥 is not free in Ⅎ𝑥𝜑. (Contributed by Mario Carneiro, 11-Aug-2016.) Remove dependency on ax-12 2171. (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 2147	. . 3 ⊢ Ⅎ𝑥∃𝑥𝜑
3		nfa1 2148	. . 3 ⊢ Ⅎ𝑥∀𝑥𝜑
4	2, 3	nfim 1899	. 2 ⊢ Ⅎ𝑥(∃𝑥𝜑 → ∀𝑥𝜑)
5	1, 4	nfxfr 1855	1 ⊢ Ⅎ𝑥Ⅎ𝑥𝜑

Syntax hints:   → wi 4  ∀wal 1539  ∃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 2137

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-ex 1782  df-nf 1786

This theorem is referenced by:  nfsb4t  2498  nfnfc1  2906  nfabdwOLD  2927  sbcnestgfw  4417  sbcnestgf  4422  bj-sbf4  35706  wl-equsal1t  36398  wl-sb6rft  36401  wl-sb8t  36405  wl-mo2tf  36424  wl-eutf  36426  wl-mo2t  36428  wl-mo3t  36429  wl-sb8eut  36430  ichnfimlem  46117  ichnfim  46118