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Theorem xfree 29629
Description: A partial converse to 19.9t 2238. (Contributed by Stefan Allan, 21-Dec-2008.) (Revised by Mario Carneiro, 11-Dec-2016.)
Assertion
Ref Expression
xfree (∀𝑥(𝜑 → ∀𝑥𝜑) ↔ ∀𝑥(∃𝑥𝜑𝜑))

Proof of Theorem xfree
StepHypRef Expression
1 nf5 2291 . 2 (Ⅎ𝑥𝜑 ↔ ∀𝑥(𝜑 → ∀𝑥𝜑))
2 nf6 2292 . 2 (Ⅎ𝑥𝜑 ↔ ∀𝑥(∃𝑥𝜑𝜑))
31, 2bitr3i 268 1 (∀𝑥(𝜑 → ∀𝑥𝜑) ↔ ∀𝑥(∃𝑥𝜑𝜑))
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 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:  xfree2  29630
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