Theorem nfnf1 2189

Description: The setvar 𝑥 is not free in Ⅎ𝑥𝜑. (Contributed by Mario Carneiro, 11-Aug-2016.) Remove dependency on ax-12 2213. (Revised by Wolf Lammen, 12-Oct-2021.)

Assertion

Ref	Expression
nfnf1	⊢ Ⅎ𝑥Ⅎ𝑥𝜑

Proof of Theorem nfnf1

Step	Hyp	Ref	Expression
1		df-nf 1814	. 2 ⊢ (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
2		nfe1 2185	. . 3 ⊢ Ⅎ𝑥∃𝑥𝜑
3		nfa1 2186	. . 3 ⊢ Ⅎ𝑥∀𝑥𝜑
4	2, 3	nfim 1926	. 2 ⊢ Ⅎ𝑥(∃𝑥𝜑 → ∀𝑥𝜑)
5	1, 4	nfxfr 1883	1 ⊢ Ⅎ𝑥Ⅎ𝑥𝜑

This proof depends on syntax axioms:   → wi 4  ∀wal 1568  ∃wex 1809  Ⅎwnf 1813

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-10 2176

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-ex 1810  df-nf 1814

This theorem is used by:  nfsb4t  2531  nfnfc1  2928  sbcnestgfw  4386  sbcnestgf  4391  bj-sbf4  37503  wl-equsal1t  38225  wl-sbid2ft  38228  wl-sb8t  38235  wl-mo2tf  38254  wl-eutf  38256  wl-mo2t  38258  wl-mo3t  38259  wl-sb8eut  38261  wl-sb8eutv  38262  ichnfimlem  48240  ichnfim  48241