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Theorem 19.36imv 1951
Description: One direction of 19.36v 1998 that can be proven without ax-6 1974. (Contributed by Rohan Ridenour, 16-Apr-2022.) (Proof shortened by Wolf Lammen, 22-Sep-2024.)
Assertion
Ref Expression
19.36imv (∃𝑥(𝜑𝜓) → (∀𝑥𝜑𝜓))
Distinct variable group:   𝜓,𝑥
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem 19.36imv
StepHypRef Expression
1 pm2.27 42 . . 3 (𝜑 → ((𝜑𝜓) → 𝜓))
21aleximi 1838 . 2 (∀𝑥𝜑 → (∃𝑥(𝜑𝜓) → ∃𝑥𝜓))
3 ax5e 1918 . 2 (∃𝑥𝜓𝜓)
42, 3syl6com 37 1 (∃𝑥(𝜑𝜓) → (∀𝑥𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1540  wex 1786
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916
This theorem depends on definitions:  df-bi 210  df-ex 1787
This theorem is referenced by:  19.36iv  1953
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