Theorem nf5 2323

Description: Alternate definition of df-nf 1811. (Contributed by Mario Carneiro, 11-Aug-2016.) df-nf 1811 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 1811	. 2 ⊢ (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
2		nfa1 2192	. . 3 ⊢ Ⅎ𝑥∀𝑥𝜑
3	2	19.23 2253	. 2 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
4	1, 3	bitr4i 281	1 ⊢ (Ⅎ𝑥𝜑 ↔ ∀𝑥(𝜑 → ∀𝑥𝜑))

Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1565  ∃wex 1806  Ⅎwnf 1810

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-10 2182  ax-12 2219

This theorem depends on definitions:  df-bi 210  df-or 861  df-ex 1807  df-nf 1811

This theorem is referenced by:  drnf1  2481  axie2  2736  xfree  32736  bj-nfdt0  37208  bj-nfalt  37226  bj-nfext  37227  bj-nfs1t  37313  wl-sbnf1  38097  hbexg  45156