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Theorem nf5 2154
Description: Alternate definition of df-nf 1750. (Contributed by Mario Carneiro, 11-Aug-2016.) df-nf 1750 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 1750 . 2 (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
2 nfa1 2068 . . 3 𝑥𝑥𝜑
3219.23 2118 . 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 1521  wex 1744  wnf 1748
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-10 2059  ax-12 2087
This theorem depends on definitions:  df-bi 197  df-or 384  df-ex 1745  df-nf 1750
This theorem is referenced by:  nfnf1OLD  2197  drnf1  2360  axie2  2626  xfree  29431  bj-nfdt0  32810  bj-nfalt  32827  bj-nfext  32828  bj-nfs1t  32839  bj-drnf1v  32875  bj-sbnf  32953  wl-sbnf1  33466  hbexg  39089
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