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Theorem bj-hbyfrbi 35026
Description: Version of bj-hbxfrbi 35025 with existential quantifiers. (Contributed by BJ, 23-Aug-2023.)
Assertion
Ref Expression
bj-hbyfrbi (((𝜑𝜓) ∧ ∀𝑥(𝜑𝜓)) → ((∃𝑥𝜑𝜑) ↔ (∃𝑥𝜓𝜓)))

Proof of Theorem bj-hbyfrbi
StepHypRef Expression
1 exbi 1849 . . 3 (∀𝑥(𝜑𝜓) → (∃𝑥𝜑 ↔ ∃𝑥𝜓))
21adantl 482 . 2 (((𝜑𝜓) ∧ ∀𝑥(𝜑𝜓)) → (∃𝑥𝜑 ↔ ∃𝑥𝜓))
3 simpl 483 . 2 (((𝜑𝜓) ∧ ∀𝑥(𝜑𝜓)) → (𝜑𝜓))
42, 3imbi12d 344 1 (((𝜑𝜓) ∧ ∀𝑥(𝜑𝜓)) → ((∃𝑥𝜑𝜑) ↔ (∃𝑥𝜓𝜓)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  wal 1539  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
This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1782
This theorem is referenced by:  bj-nnfbi  35121
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