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Theorem nf2 1603
Description: An alternate definition of df-nf 1395, 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 1395 . 2 (Ⅎ𝑥𝜑 ↔ ∀𝑥(𝜑 → ∀𝑥𝜑))
2 nfa1 1479 . . . 4 𝑥𝑥𝜑
32nfri 1457 . . 3 (∀𝑥𝜑 → ∀𝑥𝑥𝜑)
4319.23h 1432 . 2 (∀𝑥(𝜑 → ∀𝑥𝜑) ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
51, 4bitri 182 1 (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103  wal 1287  wnf 1394  wex 1426
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-gen 1383  ax-ie2 1428  ax-4 1445  ax-ial 1472
This theorem depends on definitions:  df-bi 115  df-nf 1395
This theorem is referenced by:  nf3  1604  nf4dc  1605  nf4r  1606  eusv2i  4277
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