Theorem nfnf1 2190

Description: The setvar 𝑥 is not free in Ⅎ𝑥𝜑. (Contributed by Mario Carneiro, 11-Aug-2016.) Remove dependency on ax-12 2212. (Revised by Wolf Lammen, 12-Oct-2021.)

Assertion

Ref	Expression
nfnf1	⊢ Ⅎ𝑥Ⅎ𝑥𝜑

Proof of Theorem nfnf1

Step	Hyp	Ref	Expression
1		df-nf 1816	. 2 ⊢ (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
2		nfe1 2186	. . 3 ⊢ Ⅎ𝑥∃𝑥𝜑
3		nfa1 2187	. . 3 ⊢ Ⅎ𝑥∀𝑥𝜑
4	2, 3	nfim 1928	. 2 ⊢ Ⅎ𝑥(∃𝑥𝜑 → ∀𝑥𝜑)
5	1, 4	nfxfr 1885	1 ⊢ Ⅎ𝑥Ⅎ𝑥𝜑

This proof depends on syntax axioms:   → wi 4  ∀wal 1567  ∃wex 1811  Ⅎwnf 1815

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-10 2177

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ex 1812  df-nf 1816

This theorem is used by:  nfsb4t  2528  nfnfc1  2925  sbcnestgfw  4378  sbcnestgf  4383  bj-sbf4  37752  wl-equsal1t  38475  wl-sbid2ft  38478  wl-sb8t  38485  wl-mo2tf  38504  wl-eutf  38506  wl-mo2t  38508  wl-mo3t  38509  wl-sb8eut  38511  wl-sb8eutv  38512  ichnfimlem  48545  ichnfim  48546