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Theorem xfree2 29153
Description: A partial converse to 19.9t 2069. (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 29152 . 2 (∀𝑥(𝜑 → ∀𝑥𝜑) ↔ ∀𝑥(∃𝑥𝜑𝜑))
2 eximal 1704 . . 3 ((∃𝑥𝜑𝜑) ↔ (¬ 𝜑 → ∀𝑥 ¬ 𝜑))
32albii 1744 . 2 (∀𝑥(∃𝑥𝜑𝜑) ↔ ∀𝑥𝜑 → ∀𝑥 ¬ 𝜑))
41, 3bitri 264 1 (∀𝑥(𝜑 → ∀𝑥𝜑) ↔ ∀𝑥𝜑 → ∀𝑥 ¬ 𝜑))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wal 1478  wex 1701
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-10 2016  ax-12 2044
This theorem depends on definitions:  df-bi 197  df-or 385  df-ex 1702  df-nf 1707
This theorem is referenced by: (None)
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