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Theorem nf5 2115
 Description: Alternate definition of df-nf 1709. (Contributed by Mario Carneiro, 11-Aug-2016.) df-nf 1709 changed. (Revised by Wolf Lammen, 11-Sep-2021.)
Assertion
Ref Expression
nf5 (Ⅎ𝑥𝜑 ↔ ∀𝑥(𝜑 → ∀𝑥𝜑))

Proof of Theorem nf5
StepHypRef Expression
1 df-nf 1709 . 2 (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
2 nfa1 2027 . . 3 𝑥𝑥𝜑
3219.23 2079 . 2 (∀𝑥(𝜑 → ∀𝑥𝜑) ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
41, 3bitr4i 267 1 (Ⅎ𝑥𝜑 ↔ ∀𝑥(𝜑 → ∀𝑥𝜑))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196  ∀wal 1480  ∃wex 1703  Ⅎwnf 1707 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-10 2018  ax-12 2046 This theorem depends on definitions:  df-bi 197  df-or 385  df-ex 1704  df-nf 1709 This theorem is referenced by:  nfnf1OLD  2158  drnf1  2328  axie2  2596  xfree  29287  bj-nfdt0  32669  bj-nfalt  32686  bj-nfext  32687  bj-nfs1t  32698  bj-drnf1v  32734  bj-sbnf  32812  wl-sbnf1  33316  hbexg  38598
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