Theorem nf5 2278

Description: Alternate definition of df-nf 1785. (Contributed by Mario Carneiro, 11-Aug-2016.) df-nf 1785 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 1785	. 2 ⊢ (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
2		nfa1 2147	. . 3 ⊢ Ⅎ𝑥∀𝑥𝜑
3	2	19.23 2203	. 2 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
4	1, 3	bitr4i 277	1 ⊢ (Ⅎ𝑥𝜑 ↔ ∀𝑥(𝜑 → ∀𝑥𝜑))

Syntax hints:   → wi 4   ↔ wb 205  ∀wal 1538  ∃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-5 1912  ax-6 1970  ax-7 2010  ax-10 2136  ax-12 2170

This theorem depends on definitions:  df-bi 206  df-or 845  df-ex 1781  df-nf 1785

This theorem is referenced by:  drnf1vOLD  2370  drnf1  2442  axie2  2703  xfree  30942  bj-nfdt0  34951  bj-nfalt  34967  bj-nfext  34968  bj-nfs1t  35046  bj-sbnf  35097  wl-sbnf1  35782  hbexg  42410