Theorem nfnf1 2157

Description: The setvar 𝑥 is not free in Ⅎ𝑥𝜑. (Contributed by Mario Carneiro, 11-Aug-2016.) Remove dependency on ax-12 2176. (Revised by Wolf Lammen, 12-Oct-2021.)

Assertion

Ref	Expression
nfnf1	⊢ Ⅎ𝑥Ⅎ𝑥𝜑

Proof of Theorem nfnf1

Step	Hyp	Ref	Expression
1		df-nf 1784	. 2 ⊢ (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
2		nfe1 2153	. . 3 ⊢ Ⅎ𝑥∃𝑥𝜑
3		nfa1 2154	. . 3 ⊢ Ⅎ𝑥∀𝑥𝜑
4	2, 3	nfim 1896	. 2 ⊢ Ⅎ𝑥(∃𝑥𝜑 → ∀𝑥𝜑)
5	1, 4	nfxfr 1852	1 ⊢ Ⅎ𝑥Ⅎ𝑥𝜑

Syntax hints:   → wi 4  ∀wal 1534  ∃wex 1779  Ⅎwnf 1783

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-10 2144

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ex 1780  df-nf 1784

This theorem is referenced by:  sb4bOLD  2499  nfsb4t  2538  nfsb4tALT  2603  nfnfc1  2983  nfabdw  3003  sbcnestgfw  4373  sbcnestgf  4378  bj-sbf4  34167  wl-equsal1t  34785  wl-sb6rft  34788  wl-sb8t  34792  wl-mo2tf  34811  wl-eutf  34813  wl-mo2t  34815  wl-mo3t  34816  wl-sb8eut  34817  ichnfim  43631