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Theorem 19.37imv 1948
Description: One direction of 19.37v 1998 that can be proven without ax-6 1970. (Contributed by Rohan Ridenour, 16-Apr-2022.)
Assertion
Ref Expression
19.37imv (∃𝑥(𝜑𝜓) → (𝜑 → ∃𝑥𝜓))
Distinct variable group:   𝜑,𝑥
Allowed substitution hint:   𝜓(𝑥)

Proof of Theorem 19.37imv
StepHypRef Expression
1 ax-5 1911 . 2 (𝜑 → ∀𝑥𝜑)
2 19.35 1878 . . 3 (∃𝑥(𝜑𝜓) ↔ (∀𝑥𝜑 → ∃𝑥𝜓))
32biimpi 219 . 2 (∃𝑥(𝜑𝜓) → (∀𝑥𝜑 → ∃𝑥𝜓))
41, 3syl5 34 1 (∃𝑥(𝜑𝜓) → (𝜑 → ∃𝑥𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1536  wex 1781
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911
This theorem depends on definitions:  df-bi 210  df-ex 1782
This theorem is referenced by:  19.37iv  1949
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