Theorem nf5 2315

Description: Alternate definition of df-nf 1816. (Contributed by Mario Carneiro, 11-Aug-2016.) df-nf 1816 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 1816	. 2 ⊢ (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
2		nfa1 2187	. . 3 ⊢ Ⅎ𝑥∀𝑥𝜑
3	2	19.23 2247	. 2 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
4	1, 3	bitr4i 281	1 ⊢ (Ⅎ𝑥𝜑 ↔ ∀𝑥(𝜑 → ∀𝑥𝜑))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209  ∀wal 1567  ∃wex 1811  Ⅎwnf 1815

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-10 2177  ax-12 2212

This proof depends on definitions:  df-bi 210  df-or 862  df-ex 1812  df-nf 1816

This theorem is used by:  drnf1  2472  axie2  2727  xfree  33097  bj-nfdt0  37721  bj-nfalt  37739  bj-nfext  37740  bj-nfs1t  37826  wl-sbnf1  38624  hbexg  45675