Theorem nf5 2282

Description: Alternate definition of df-nf 1784. (Contributed by Mario Carneiro, 11-Aug-2016.) df-nf 1784 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 1784	. 2 ⊢ (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
2		nfa1 2151	. . 3 ⊢ Ⅎ𝑥∀𝑥𝜑
3	2	19.23 2211	. 2 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
4	1, 3	bitr4i 278	1 ⊢ (Ⅎ𝑥𝜑 ↔ ∀𝑥(𝜑 → ∀𝑥𝜑))

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1538  ∃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-5 1910  ax-6 1967  ax-7 2007  ax-10 2141  ax-12 2177

This theorem depends on definitions:  df-bi 207  df-or 849  df-ex 1780  df-nf 1784

This theorem is referenced by:  sbnfOLD  2313  drnf1vOLD  2376  drnf1  2448  axie2  2703  xfree  32463  bj-nfdt0  36696  bj-nfalt  36712  bj-nfext  36713  bj-nfs1t  36791  wl-sbnf1  37556  hbexg  44576