Theorem nfnf1 2151

Description: The setvar 𝑥 is not free in Ⅎ𝑥𝜑. (Contributed by Mario Carneiro, 11-Aug-2016.) Remove dependency on ax-12 2171. (Revised by Wolf Lammen, 12-Oct-2021.)

Assertion

Ref	Expression
nfnf1	⊢ Ⅎ𝑥Ⅎ𝑥𝜑

Proof of Theorem nfnf1

Step	Hyp	Ref	Expression
1		df-nf 1787	. 2 ⊢ (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
2		nfe1 2147	. . 3 ⊢ Ⅎ𝑥∃𝑥𝜑
3		nfa1 2148	. . 3 ⊢ Ⅎ𝑥∀𝑥𝜑
4	2, 3	nfim 1899	. 2 ⊢ Ⅎ𝑥(∃𝑥𝜑 → ∀𝑥𝜑)
5	1, 4	nfxfr 1855	1 ⊢ Ⅎ𝑥Ⅎ𝑥𝜑

Syntax hints:   → wi 4  ∀wal 1537  ∃wex 1782  Ⅎwnf 1786

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-10 2137

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-ex 1783  df-nf 1787

This theorem is referenced by:  sb4bOLD  2476  nfsb4t  2503  nfnfc1  2910  nfabdwOLD  2931  sbcnestgfw  4352  sbcnestgf  4357  bj-sbf4  35023  wl-equsal1t  35700  wl-sb6rft  35703  wl-sb8t  35707  wl-mo2tf  35726  wl-eutf  35728  wl-mo2t  35730  wl-mo3t  35731  wl-sb8eut  35732  ichnfimlem  44915  ichnfim  44916