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Theorem xfree2 29630
Description: A partial converse to 19.9t 2238. (Contributed by Stefan Allan, 21-Dec-2008.)
Assertion
Ref Expression
xfree2 (∀𝑥(𝜑 → ∀𝑥𝜑) ↔ ∀𝑥𝜑 → ∀𝑥 ¬ 𝜑))

Proof of Theorem xfree2
StepHypRef Expression
1 xfree 29629 . 2 (∀𝑥(𝜑 → ∀𝑥𝜑) ↔ ∀𝑥(∃𝑥𝜑𝜑))
2 eximal 1862 . . 3 ((∃𝑥𝜑𝜑) ↔ (¬ 𝜑 → ∀𝑥 ¬ 𝜑))
32albii 1904 . 2 (∀𝑥(∃𝑥𝜑𝜑) ↔ ∀𝑥𝜑 → ∀𝑥 ¬ 𝜑))
41, 3bitri 266 1 (∀𝑥(𝜑 → ∀𝑥𝜑) ↔ ∀𝑥𝜑 → ∀𝑥 ¬ 𝜑))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wal 1635  wex 1859
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 2068  ax-7 2104  ax-10 2185  ax-12 2214
This theorem depends on definitions:  df-bi 198  df-or 866  df-ex 1860  df-nf 1864
This theorem is referenced by: (None)
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