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Theorem 19.9ht 1605
Description: A closed version of one direction of 19.9 1608. (Contributed by NM, 5-Aug-1993.)
Assertion
Ref Expression
19.9ht (∀𝑥(𝜑 → ∀𝑥𝜑) → (∃𝑥𝜑𝜑))

Proof of Theorem 19.9ht
StepHypRef Expression
1 id 19 . . 3 (𝜑𝜑)
21ax-gen 1410 . 2 𝑥(𝜑𝜑)
3 19.23ht 1458 . 2 (∀𝑥(𝜑 → ∀𝑥𝜑) → (∀𝑥(𝜑𝜑) ↔ (∃𝑥𝜑𝜑)))
42, 3mpbii 147 1 (∀𝑥(𝜑 → ∀𝑥𝜑) → (∃𝑥𝜑𝜑))
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1314  wex 1453
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-gen 1410  ax-ie2 1455
This theorem depends on definitions:  df-bi 116
This theorem is referenced by:  19.9t  1606  19.9h  1607  19.9hd  1625
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