Theorem nfnf1 2018

Description: The setvar 𝑥 is not free in Ⅎ𝑥𝜑. (Contributed by Mario Carneiro, 11-Aug-2016.) Remove dependency on ax-12 2034. (Revised by Wolf Lammen, 12-Oct-2021.)

Assertion

Ref	Expression
nfnf1	⊢ Ⅎ𝑥Ⅎ𝑥𝜑

Proof of Theorem nfnf1

Step	Hyp	Ref	Expression
1		df-nf 1701	. 2 ⊢ (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
2		nfe1 2014	. . 3 ⊢ Ⅎ𝑥∃𝑥𝜑
3		nfa1 2015	. . 3 ⊢ Ⅎ𝑥∀𝑥𝜑
4	2, 3	nfim 1813	. 2 ⊢ Ⅎ𝑥(∃𝑥𝜑 → ∀𝑥𝜑)
5	1, 4	nfxfr 1771	1 ⊢ Ⅎ𝑥Ⅎ𝑥𝜑

Syntax hints:   → wi 4  ∀wal 1473  ∃wex 1695  Ⅎwnf 1699

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-10 2006

This theorem depends on definitions:  df-bi 196  df-or 384  df-tru 1478  df-ex 1696  df-nf 1701

This theorem is referenced by:  nfaldOLD  2152  nfeqf2  2285  nfsb4t  2377  nfnfc1  2754  sbcnestgf  3947  dfnfc2OLD  4391  bj-sbf4  32015  wl-equsal1t  32506  wl-sb6rft  32509  wl-sb8t  32512  wl-mo2tf  32532  wl-eutf  32534  wl-mo2t  32536  wl-mo3t  32537  wl-sb8eut  32538