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Theorem bj-19.42t 34178
Description: Closed form of 19.42 2239 from the same axioms as 19.42v 1954. (Contributed by BJ, 2-Dec-2023.)
Assertion
Ref Expression
bj-19.42t (Ⅎ'𝑥𝜑 → (∃𝑥(𝜑𝜓) ↔ (𝜑 ∧ ∃𝑥𝜓)))

Proof of Theorem bj-19.42t
StepHypRef Expression
1 19.40 1887 . . 3 (∃𝑥(𝜑𝜓) → (∃𝑥𝜑 ∧ ∃𝑥𝜓))
2 bj-nnfe 34138 . . . 4 (Ⅎ'𝑥𝜑 → (∃𝑥𝜑𝜑))
32anim1d 613 . . 3 (Ⅎ'𝑥𝜑 → ((∃𝑥𝜑 ∧ ∃𝑥𝜓) → (𝜑 ∧ ∃𝑥𝜓)))
41, 3syl5 34 . 2 (Ⅎ'𝑥𝜑 → (∃𝑥(𝜑𝜓) → (𝜑 ∧ ∃𝑥𝜓)))
5 bj-nnfa 34136 . . . 4 (Ⅎ'𝑥𝜑 → (𝜑 → ∀𝑥𝜑))
65anim1d 613 . . 3 (Ⅎ'𝑥𝜑 → ((𝜑 ∧ ∃𝑥𝜓) → (∀𝑥𝜑 ∧ ∃𝑥𝜓)))
7 19.29 1874 . . 3 ((∀𝑥𝜑 ∧ ∃𝑥𝜓) → ∃𝑥(𝜑𝜓))
86, 7syl6 35 . 2 (Ⅎ'𝑥𝜑 → ((𝜑 ∧ ∃𝑥𝜓) → ∃𝑥(𝜑𝜓)))
94, 8impbid 215 1 (Ⅎ'𝑥𝜑 → (∃𝑥(𝜑𝜓) ↔ (𝜑 ∧ ∃𝑥𝜓)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wal 1536  wex 1781  Ⅎ'wnnf 34131
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 34132
This theorem is referenced by:  bj-19.41t  34179
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