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Theorem nf5 2288
Description: Alternate definition of df-nf 1786. (Contributed by Mario Carneiro, 11-Aug-2016.) df-nf 1786 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 1786 . 2 (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
2 nfa1 2153 . . 3 𝑥𝑥𝜑
3219.23 2210 . 2 (∀𝑥(𝜑 → ∀𝑥𝜑) ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
41, 3bitr4i 281 1 (Ⅎ𝑥𝜑 ↔ ∀𝑥(𝜑 → ∀𝑥𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wal 1536  wex 1781  wnf 1785
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-10 2143  ax-12 2176
This theorem depends on definitions:  df-bi 210  df-or 845  df-ex 1782  df-nf 1786
This theorem is referenced by:  drnf1v  2381  drnf1  2457  axie2  2768  xfree  30231  bj-nfdt0  34143  bj-nfalt  34159  bj-nfext  34160  bj-nfs1t  34228  bj-sbnf  34280  wl-sbnf1  34955  hbexg  41259
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