Theorem nfnf1 2155

Description: The setvar 𝑥 is not free in Ⅎ𝑥𝜑. (Contributed by Mario Carneiro, 11-Aug-2016.) Remove dependency on ax-12 2178. (Revised by Wolf Lammen, 12-Oct-2021.)

Assertion

Ref	Expression
nfnf1	⊢ Ⅎ𝑥Ⅎ𝑥𝜑

Proof of Theorem nfnf1

Step	Hyp	Ref	Expression
1		df-nf 1782	. 2 ⊢ (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
2		nfe1 2151	. . 3 ⊢ Ⅎ𝑥∃𝑥𝜑
3		nfa1 2152	. . 3 ⊢ Ⅎ𝑥∀𝑥𝜑
4	2, 3	nfim 1895	. 2 ⊢ Ⅎ𝑥(∃𝑥𝜑 → ∀𝑥𝜑)
5	1, 4	nfxfr 1851	1 ⊢ Ⅎ𝑥Ⅎ𝑥𝜑

Syntax hints:   → wi 4  ∀wal 1535  ∃wex 1777  Ⅎwnf 1781

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-10 2141

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-ex 1778  df-nf 1782

This theorem is referenced by:  nfsb4t  2507  nfnfc1  2911  nfabdwOLD  2933  sbcnestgfw  4444  sbcnestgf  4449  bj-sbf4  36806  wl-equsal1t  37496  wl-sbid2ft  37499  wl-sb8t  37506  wl-mo2tf  37525  wl-eutf  37527  wl-mo2t  37529  wl-mo3t  37530  wl-sb8eut  37532  wl-sb8eutv  37533  ichnfimlem  47337  ichnfim  47338