Theorem nf5 2316

Description: Alternate definition of df-nf 1804. (Contributed by Mario Carneiro, 11-Aug-2016.) df-nf 1804 changed. (Revised by Wolf Lammen, 11-Sep-2021.)

Assertion

Ref	Expression
nf5	⊢ (Ⅎ𝑥𝜑 ↔ ∀𝑥(𝜑 → ∀𝑥𝜑))

Proof of Theorem nf5

Step	Hyp	Ref	Expression
1		df-nf 1804	. 2 ⊢ (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
2		nfa1 2185	. . 3 ⊢ Ⅎ𝑥∀𝑥𝜑
3	2	19.23 2246	. 2 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
4	1, 3	bitr4i 280	1 ⊢ (Ⅎ𝑥𝜑 ↔ ∀𝑥(𝜑 → ∀𝑥𝜑))

Syntax hints:   → wi 4   ↔ wb 208  ∀wal 1558  ∃wex 1799  Ⅎwnf 1803

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-10 2175  ax-12 2212

This theorem depends on definitions:  df-bi 209  df-or 859  df-ex 1800  df-nf 1804

This theorem is referenced by:  drnf1  2474  axie2  2729  xfree  32644  bj-nfdt0  37167  bj-nfalt  37185  bj-nfext  37186  bj-nfs1t  37272  wl-sbnf1  38055  hbexg  45129