Theorem nf2 1558

Description: An alternative definition of df-nf 1350, 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 1350	. 2 ⊢ (Ⅎ𝑥𝜑 ↔ ∀𝑥(𝜑 → ∀𝑥𝜑))
2		nfa1 1434	. . . 4 ⊢ Ⅎ𝑥∀𝑥𝜑
3	2	nfri 1412	. . 3 ⊢ (∀𝑥𝜑 → ∀𝑥∀𝑥𝜑)
4	3	19.23h 1387	. 2 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
5	1, 4	bitri 173	1 ⊢ (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))

Syntax hints:   → wi 4   ↔ wb 98  ∀wal 1241  Ⅎwnf 1349  ∃wex 1381

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-gen 1338  ax-ie2 1383  ax-4 1400  ax-ial 1427

This theorem depends on definitions:  df-bi 110  df-nf 1350

This theorem is referenced by:  nf3  1559  nf4dc  1560  nf4r  1561  eusv2i  4187