Theorem nfnf1 2159

Description: The setvar 𝑥 is not free in Ⅎ𝑥𝜑. (Contributed by Mario Carneiro, 11-Aug-2016.) Remove dependency on ax-12 2184. (Revised by Wolf Lammen, 12-Oct-2021.)

Assertion

Ref	Expression
nfnf1	⊢ Ⅎ𝑥Ⅎ𝑥𝜑

Proof of Theorem nfnf1

Step	Hyp	Ref	Expression
1		df-nf 1785	. 2 ⊢ (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
2		nfe1 2155	. . 3 ⊢ Ⅎ𝑥∃𝑥𝜑
3		nfa1 2156	. . 3 ⊢ Ⅎ𝑥∀𝑥𝜑
4	2, 3	nfim 1897	. 2 ⊢ Ⅎ𝑥(∃𝑥𝜑 → ∀𝑥𝜑)
5	1, 4	nfxfr 1854	1 ⊢ Ⅎ𝑥Ⅎ𝑥𝜑

Syntax hints:   → wi 4  ∀wal 1539  ∃wex 1780  Ⅎwnf 1784

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-10 2146

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ex 1781  df-nf 1785

This theorem is referenced by:  nfsb4t  2503  nfnfc1  2901  sbcnestgfw  4373  sbcnestgf  4378  bj-sbf4  37043  wl-equsal1t  37749  wl-sbid2ft  37752  wl-sb8t  37759  wl-mo2tf  37778  wl-eutf  37780  wl-mo2t  37782  wl-mo3t  37783  wl-sb8eut  37785  wl-sb8eutv  37786  ichnfimlem  47730  ichnfim  47731