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Theorem 19.36imv 2041
Description: One direction of 19.36v 2087 that can be proven without ax-6 2072. (Contributed by Rohan Ridenour, 16-Apr-2022.)
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 19.35 1977 . . 3 (∃𝑥(𝜑𝜓) ↔ (∀𝑥𝜑 → ∃𝑥𝜓))
21biimpi 208 . 2 (∃𝑥(𝜑𝜓) → (∀𝑥𝜑 → ∃𝑥𝜓))
3 ax5e 2008 . 2 (∃𝑥𝜓𝜓)
42, 3syl6 35 1 (∃𝑥(𝜑𝜓) → (∀𝑥𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1651  wex 1875
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006
This theorem depends on definitions:  df-bi 199  df-ex 1876
This theorem is referenced by:  19.36iv  2042
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