Theorem nfnf1 2196

Description: The setvar 𝑥 is not free in Ⅎ𝑥𝜑. (Contributed by Mario Carneiro, 11-Aug-2016.) Remove dependency on ax-12 2211. (Revised by Wolf Lammen, 12-Oct-2021.)

Assertion

Ref	Expression
nfnf1	⊢ Ⅎ𝑥Ⅎ𝑥𝜑

Proof of Theorem nfnf1

Step	Hyp	Ref	Expression
1		df-nf 1879	. 2 ⊢ (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
2		nfe1 2192	. . 3 ⊢ Ⅎ𝑥∃𝑥𝜑
3		nfa1 2193	. . 3 ⊢ Ⅎ𝑥∀𝑥𝜑
4	2, 3	nfim 1995	. 2 ⊢ Ⅎ𝑥(∃𝑥𝜑 → ∀𝑥𝜑)
5	1, 4	nfxfr 1948	1 ⊢ Ⅎ𝑥Ⅎ𝑥𝜑

Syntax hints:   → wi 4  ∀wal 1650  ∃wex 1874  Ⅎwnf 1878

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-10 2183

This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-ex 1875  df-nf 1879

This theorem is referenced by:  nfeqf2OLD  2397  nfsb4t  2480  nfnfc1  2910  sbcnestgf  4158  bj-sbf4  33259  wl-equsal1t  33755  wl-sb6rft  33759  wl-sb8t  33762  wl-mo2tf  33781  wl-eutf  33783  wl-mo2t  33785  wl-mo3t  33786  wl-sb8eut  33787