Theorem nf2 1627

Description: An alternate definition of df-nf 1418, 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 1418	. 2 ⊢ (Ⅎ𝑥𝜑 ↔ ∀𝑥(𝜑 → ∀𝑥𝜑))
2		nfa1 1502	. . . 4 ⊢ Ⅎ𝑥∀𝑥𝜑
3	2	nfri 1480	. . 3 ⊢ (∀𝑥𝜑 → ∀𝑥∀𝑥𝜑)
4	3	19.23h 1455	. 2 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
5	1, 4	bitri 183	1 ⊢ (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))

Syntax hints:   → wi 4   ↔ wb 104  ∀wal 1310  Ⅎwnf 1417  ∃wex 1449

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-gen 1406  ax-ie2 1451  ax-4 1468  ax-ial 1495

This theorem depends on definitions:  df-bi 116  df-nf 1418

This theorem is referenced by:  nf3  1628  nf4dc  1629  nf4r  1630  eusv2i  4334