Theorem nfnf1 2190

Description: The setvar 𝑥 is not free in Ⅎ𝑥𝜑. (Contributed by Mario Carneiro, 11-Aug-2016.) Remove dependency on ax-12 2214. (Revised by Wolf Lammen, 12-Oct-2021.)

Assertion

Ref	Expression
nfnf1	⊢ Ⅎ𝑥Ⅎ𝑥𝜑

Proof of Theorem nfnf1

Step	Hyp	Ref	Expression
1		df-nf 1806	. 2 ⊢ (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
2		nfe1 2186	. . 3 ⊢ Ⅎ𝑥∃𝑥𝜑
3		nfa1 2187	. . 3 ⊢ Ⅎ𝑥∀𝑥𝜑
4	2, 3	nfim 1918	. 2 ⊢ Ⅎ𝑥(∃𝑥𝜑 → ∀𝑥𝜑)
5	1, 4	nfxfr 1875	1 ⊢ Ⅎ𝑥Ⅎ𝑥𝜑

Syntax hints:   → wi 4  ∀wal 1560  ∃wex 1801  Ⅎwnf 1805

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-10 2177

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-ex 1802  df-nf 1806

This theorem is referenced by:  nfsb4t  2532  nfnfc1  2929  sbcnestgfw  4377  sbcnestgf  4382  bj-sbf4  37330  wl-equsal1t  38050  wl-sbid2ft  38053  wl-sb8t  38060  wl-mo2tf  38079  wl-eutf  38081  wl-mo2t  38083  wl-mo3t  38084  wl-sb8eut  38086  wl-sb8eutv  38087  ichnfimlem  48074  ichnfim  48075