Theorem nf5 2279

Description: Alternate definition of df-nf 1787. (Contributed by Mario Carneiro, 11-Aug-2016.) df-nf 1787 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 1787	. 2 ⊢ (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
2		nfa1 2148	. . 3 ⊢ Ⅎ𝑥∀𝑥𝜑
3	2	19.23 2204	. 2 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
4	1, 3	bitr4i 277	1 ⊢ (Ⅎ𝑥𝜑 ↔ ∀𝑥(𝜑 → ∀𝑥𝜑))

Syntax hints:   → wi 4   ↔ wb 205  ∀wal 1537  ∃wex 1782  Ⅎwnf 1786

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-10 2137  ax-12 2171

This theorem depends on definitions:  df-bi 206  df-or 845  df-ex 1783  df-nf 1787

This theorem is referenced by:  drnf1vOLD  2371  drnf1  2443  axie2  2704  xfree  30806  bj-nfdt0  34877  bj-nfalt  34893  bj-nfext  34894  bj-nfs1t  34972  bj-sbnf  35024  wl-sbnf1  35710  hbexg  42176