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Theorem alexbii 1750
 Description: Biconditional form of aleximi 1749. (Contributed by BJ, 16-Nov-2020.)
Hypothesis
Ref Expression
alexbii.1 (𝜑 → (𝜓𝜒))
Assertion
Ref Expression
alexbii (∀𝑥𝜑 → (∃𝑥𝜓 ↔ ∃𝑥𝜒))

Proof of Theorem alexbii
StepHypRef Expression
1 alexbii.1 . . . 4 (𝜑 → (𝜓𝜒))
21biimpd 218 . . 3 (𝜑 → (𝜓𝜒))
32aleximi 1749 . 2 (∀𝑥𝜑 → (∃𝑥𝜓 → ∃𝑥𝜒))
41biimprd 237 . . 3 (𝜑 → (𝜒𝜓))
54aleximi 1749 . 2 (∀𝑥𝜑 → (∃𝑥𝜒 → ∃𝑥𝜓))
63, 5impbid 201 1 (∀𝑥𝜑 → (∃𝑥𝜓 ↔ ∃𝑥𝜒))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195  ∀wal 1473  ∃wex 1695 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728 This theorem depends on definitions:  df-bi 196  df-ex 1696 This theorem is referenced by:  exbi  1762  exbidh  1781  exintrbi  1808  eleq2d  2673
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