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Theorem xfree 29908
Description: A partial converse to 19.9t 2171. (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 2258 . 2 (Ⅎ𝑥𝜑 ↔ ∀𝑥(𝜑 → ∀𝑥𝜑))
2 nf6 2259 . 2 (Ⅎ𝑥𝜑 ↔ ∀𝑥(∃𝑥𝜑𝜑))
31, 2bitr3i 278 1 (∀𝑥(𝜑 → ∀𝑥𝜑) ↔ ∀𝑥(∃𝑥𝜑𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wal 1523  wex 1765  wnf 1769
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-10 2114  ax-12 2143
This theorem depends on definitions:  df-bi 208  df-or 843  df-ex 1766  df-nf 1770
This theorem is referenced by:  xfree2  29909
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