Theorem nf5 2281

Description: Alternate definition of df-nf 1781. (Contributed by Mario Carneiro, 11-Aug-2016.) df-nf 1781 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 1781	. 2 ⊢ (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
2		nfa1 2149	. . 3 ⊢ Ⅎ𝑥∀𝑥𝜑
3	2	19.23 2209	. 2 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
4	1, 3	bitr4i 278	1 ⊢ (Ⅎ𝑥𝜑 ↔ ∀𝑥(𝜑 → ∀𝑥𝜑))

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1535  ∃wex 1776  Ⅎwnf 1780

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-10 2139  ax-12 2175

This theorem depends on definitions:  df-bi 207  df-or 848  df-ex 1777  df-nf 1781

This theorem is referenced by:  sbnfOLD  2312  drnf1vOLD  2374  drnf1  2446  axie2  2701  xfree  32473  bj-nfdt0  36678  bj-nfalt  36694  bj-nfext  36695  bj-nfs1t  36773  wl-sbnf1  37536  hbexg  44554