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Theorem 19.9ht 1532
Description: A closed version of one direction of 19.9 1535. (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 1338 . 2 𝑥(𝜑𝜑)
3 19.23ht 1386 . 2 (∀𝑥(𝜑 → ∀𝑥𝜑) → (∀𝑥(𝜑𝜑) ↔ (∃𝑥𝜑𝜑)))
42, 3mpbii 136 1 (∀𝑥(𝜑 → ∀𝑥𝜑) → (∃𝑥𝜑𝜑))
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1241  wex 1381
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-gen 1338  ax-ie2 1383
This theorem depends on definitions:  df-bi 110
This theorem is referenced by:  19.9t  1533  19.9h  1534  19.9hd  1552
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