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Theorem nf5 2293
Description: Alternate definition of df-nf 1864. (Contributed by Mario Carneiro, 11-Aug-2016.) df-nf 1864 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 1864 . 2 (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
2 nfa1 2196 . . 3 𝑥𝑥𝜑
3219.23 2248 . 2 (∀𝑥(𝜑 → ∀𝑥𝜑) ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
41, 3bitr4i 269 1 (Ⅎ𝑥𝜑 ↔ ∀𝑥(𝜑 → ∀𝑥𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wal 1635  wex 1859  wnf 1863
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-10 2186  ax-12 2215
This theorem depends on definitions:  df-bi 198  df-or 866  df-ex 1860  df-nf 1864
This theorem is referenced by:  nfnf1OLD  2337  drnf1  2493  axie2  2786  xfree  29637  bj-nfdt0  33005  bj-nfalt  33022  bj-nfext  33023  bj-nfs1t  33034  bj-drnf1v  33069  bj-sbnf  33143  wl-sbnf1  33653  hbexg  39271
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