Theorem nfnf1 2031

Description: The setvar 𝑥 is not free in Ⅎ𝑥𝜑. (Contributed by Mario Carneiro, 11-Aug-2016.) Remove dependency on ax-12 2047. (Revised by Wolf Lammen, 12-Oct-2021.)

Assertion

Ref	Expression
nfnf1	⊢ Ⅎ𝑥Ⅎ𝑥𝜑

Proof of Theorem nfnf1

Step	Hyp	Ref	Expression
1		df-nf 1710	. 2 ⊢ (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
2		nfe1 2027	. . 3 ⊢ Ⅎ𝑥∃𝑥𝜑
3		nfa1 2028	. . 3 ⊢ Ⅎ𝑥∀𝑥𝜑
4	2, 3	nfim 1825	. 2 ⊢ Ⅎ𝑥(∃𝑥𝜑 → ∀𝑥𝜑)
5	1, 4	nfxfr 1779	1 ⊢ Ⅎ𝑥Ⅎ𝑥𝜑

Syntax hints:   → wi 4  ∀wal 1481  ∃wex 1704  Ⅎwnf 1708

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-10 2019

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1705  df-nf 1710

This theorem is referenced by:  nfaldOLD  2166  nfeqf2  2297  nfsb4t  2389  nfnfc1  2767  sbcnestgf  3995  dfnfc2OLD  4455  bj-sbf4  32827  wl-equsal1t  33327  wl-sb6rft  33330  wl-sb8t  33333  wl-mo2tf  33353  wl-eutf  33355  wl-mo2t  33357  wl-mo3t  33358  wl-sb8eut  33359