Theorem nf2 1866

Description: An alternative definition of df-nf 1545, which does not involve nested quantifiers on the same variable. (Contributed by Mario Carneiro, 24-Sep-2016.)

Assertion

Ref	Expression
nf2	⊢ (Ⅎxφ ↔ (∃xφ → ∀xφ))

Proof of Theorem nf2

Step	Hyp	Ref	Expression
1		df-nf 1545	. 2 ⊢ (Ⅎxφ ↔ ∀x(φ → ∀xφ))
2		nfa1 1788	. . 3 ⊢ Ⅎx∀xφ
3	2	19.23 1801	. 2 ⊢ (∀x(φ → ∀xφ) ↔ (∃xφ → ∀xφ))
4	1, 3	bitri 240	1 ⊢ (Ⅎxφ ↔ (∃xφ → ∀xφ))

Syntax hints:   → wi 4   ↔ wb 176  ∀wal 1540  ∃wex 1541  Ⅎwnf 1544

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-11 1746

This theorem depends on definitions:  df-bi 177  df-ex 1542  df-nf 1545

This theorem is referenced by:  nf3  1867  nf4  1868