Theorem nfnf1 2195

Description: The setvar 𝑥 is not free in Ⅎ𝑥𝜑. (Contributed by Mario Carneiro, 11-Aug-2016.) Remove dependency on ax-12 2219. (Revised by Wolf Lammen, 12-Oct-2021.)

Assertion

Ref	Expression
nfnf1	⊢ Ⅎ𝑥Ⅎ𝑥𝜑

Proof of Theorem nfnf1

Step	Hyp	Ref	Expression
1		df-nf 1811	. 2 ⊢ (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
2		nfe1 2191	. . 3 ⊢ Ⅎ𝑥∃𝑥𝜑
3		nfa1 2192	. . 3 ⊢ Ⅎ𝑥∀𝑥𝜑
4	2, 3	nfim 1923	. 2 ⊢ Ⅎ𝑥(∃𝑥𝜑 → ∀𝑥𝜑)
5	1, 4	nfxfr 1880	1 ⊢ Ⅎ𝑥Ⅎ𝑥𝜑

Syntax hints:   → wi 4  ∀wal 1565  ∃wex 1806  Ⅎwnf 1810

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-10 2182

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-ex 1807  df-nf 1811

This theorem is referenced by:  nfsb4t  2537  nfnfc1  2934  sbcnestgfw  4392  sbcnestgf  4397  bj-sbf4  37398  wl-equsal1t  38119  wl-sbid2ft  38122  wl-sb8t  38129  wl-mo2tf  38148  wl-eutf  38150  wl-mo2t  38152  wl-mo3t  38153  wl-sb8eut  38155  wl-sb8eutv  38156  ichnfimlem  48135  ichnfim  48136