Theorem nfnf1 2155

Description: The setvar 𝑥 is not free in Ⅎ𝑥𝜑. (Contributed by Mario Carneiro, 11-Aug-2016.) Remove dependency on ax-12 2175. (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 2151	. . 3 ⊢ Ⅎ𝑥∃𝑥𝜑
3		nfa1 2152	. . 3 ⊢ Ⅎ𝑥∀𝑥𝜑
4	2, 3	nfim 1897	. 2 ⊢ Ⅎ𝑥(∃𝑥𝜑 → ∀𝑥𝜑)
5	1, 4	nfxfr 1854	1 ⊢ Ⅎ𝑥Ⅎ𝑥𝜑

Syntax hints:   → wi 4  ∀wal 1536  ∃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 2142

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-nf 1786

This theorem is referenced by:  sb4bOLD  2489  nfsb4t  2517  nfsb4tALT  2580  nfnfc1  2958  nfabdw  2976  sbcnestgfw  4326  sbcnestgf  4331  bj-sbf4  34278  wl-equsal1t  34946  wl-sb6rft  34949  wl-sb8t  34953  wl-mo2tf  34972  wl-eutf  34974  wl-mo2t  34976  wl-mo3t  34977  wl-sb8eut  34978  ichnfimlem  43980  ichnfim  43981