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Theorem nf2 1574
 Description: An alternative definition of df-nf 1366, 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
StepHypRef Expression
1 df-nf 1366 . 2 (Ⅎ𝑥𝜑 ↔ ∀𝑥(𝜑 → ∀𝑥𝜑))
2 nfa1 1450 . . . 4 𝑥𝑥𝜑
32nfri 1428 . . 3 (∀𝑥𝜑 → ∀𝑥𝑥𝜑)
4319.23h 1403 . 2 (∀𝑥(𝜑 → ∀𝑥𝜑) ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
51, 4bitri 177 1 (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 102  ∀wal 1257  Ⅎwnf 1365  ∃wex 1397 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-gen 1354  ax-ie2 1399  ax-4 1416  ax-ial 1443 This theorem depends on definitions:  df-bi 114  df-nf 1366 This theorem is referenced by:  nf3  1575  nf4dc  1576  nf4r  1577  eusv2i  4215
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