Theorem nfnf1 2152

Description: The setvar 𝑥 is not free in Ⅎ𝑥𝜑. (Contributed by Mario Carneiro, 11-Aug-2016.) Remove dependency on ax-12 2172. (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 2148	. . 3 ⊢ Ⅎ𝑥∃𝑥𝜑
3		nfa1 2149	. . 3 ⊢ Ⅎ𝑥∀𝑥𝜑
4	2, 3	nfim 1900	. 2 ⊢ Ⅎ𝑥(∃𝑥𝜑 → ∀𝑥𝜑)
5	1, 4	nfxfr 1856	1 ⊢ Ⅎ𝑥Ⅎ𝑥𝜑

Syntax hints:   → wi 4  ∀wal 1540  ∃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 2138

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-ex 1783  df-nf 1787

This theorem is referenced by:  nfsb4t  2499  nfnfc1  2907  nfabdwOLD  2928  sbcnestgfw  4419  sbcnestgf  4424  bj-sbf4  35718  wl-equsal1t  36410  wl-sb6rft  36413  wl-sb8t  36417  wl-mo2tf  36436  wl-eutf  36438  wl-mo2t  36440  wl-mo3t  36441  wl-sb8eut  36442  ichnfimlem  46131  ichnfim  46132