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Theorem bj-19.37im 34217
Description: One direction of 19.37 2233 from the same axioms as 19.37imv 1948. (Contributed by BJ, 2-Dec-2023.)
Assertion
Ref Expression
bj-19.37im (Ⅎ'𝑥𝜑 → (∃𝑥(𝜑𝜓) → (𝜑 → ∃𝑥𝜓)))

Proof of Theorem bj-19.37im
StepHypRef Expression
1 19.35 1878 . 2 (∃𝑥(𝜑𝜓) ↔ (∀𝑥𝜑 → ∃𝑥𝜓))
2 bj-nnfa 34176 . . 3 (Ⅎ'𝑥𝜑 → (𝜑 → ∀𝑥𝜑))
32imim1d 82 . 2 (Ⅎ'𝑥𝜑 → ((∀𝑥𝜑 → ∃𝑥𝜓) → (𝜑 → ∃𝑥𝜓)))
41, 3syl5bi 245 1 (Ⅎ'𝑥𝜑 → (∃𝑥(𝜑𝜓) → (𝜑 → ∃𝑥𝜓)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1536  wex 1781  Ⅎ'wnnf 34171
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-bj-nnf 34172
This theorem is referenced by: (None)
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