Theorem nf5 2285

Description: Alternate definition of df-nf 1792. (Contributed by Mario Carneiro, 11-Aug-2016.) df-nf 1792 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 1792	. 2 ⊢ (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
2		nfa1 2154	. . 3 ⊢ Ⅎ𝑥∀𝑥𝜑
3	2	19.23 2211	. 2 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
4	1, 3	bitr4i 281	1 ⊢ (Ⅎ𝑥𝜑 ↔ ∀𝑥(𝜑 → ∀𝑥𝜑))

Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1541  ∃wex 1787  Ⅎwnf 1791

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-10 2143  ax-12 2177

This theorem depends on definitions:  df-bi 210  df-or 848  df-ex 1788  df-nf 1792

This theorem is referenced by:  drnf1v  2370  drnf1  2442  axie2  2703  xfree  30479  bj-nfdt0  34563  bj-nfalt  34579  bj-nfext  34580  bj-nfs1t  34658  bj-sbnf  34710  wl-sbnf1  35396  hbexg  41790