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Theorem 19.9ht 2020
Description: A closed version of 19.9 2023. (Contributed by NM, 13-May-1993.) (Proof shortened by Wolf Lammen, 3-Mar-2018.)
Assertion
Ref Expression
19.9ht (∀𝑥(𝜑 → ∀𝑥𝜑) → (∃𝑥𝜑𝜑))

Proof of Theorem 19.9ht
StepHypRef Expression
1 exim 1737 . 2 (∀𝑥(𝜑 → ∀𝑥𝜑) → (∃𝑥𝜑 → ∃𝑥𝑥𝜑))
2 axc7e 1993 . 2 (∃𝑥𝑥𝜑𝜑)
31, 2syl6 34 1 (∀𝑥(𝜑 → ∀𝑥𝜑) → (∃𝑥𝜑𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1466  wex 1692
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1698  ax-4 1711  ax-5 1789  ax-6 1836  ax-7 1883  ax-10 1965  ax-12 1983
This theorem depends on definitions:  df-bi 192  df-ex 1693
This theorem is referenced by:  19.9d  2021  19.9tOLD  2025  hbnt  2029  bj-19.9htbi  31482
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