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Theorem sepex 5305
Description: Convert implication to equivalence within an existence statement using the Separation Scheme (Aussonderung) ax-sep 5301. Similar to Theorem 1.3(ii) of [BellMachover] p. 463. (Contributed by Matthew House, 19-Sep-2025.)
Assertion
Ref Expression
sepex (∃𝑦𝑥(𝜑𝑥𝑦) → ∃𝑧𝑥(𝑥𝑧𝜑))
Distinct variable groups:   𝑥,𝑦   𝑥,𝑧   𝜑,𝑦   𝜑,𝑧
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem sepex
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 sepexlem 5304 . 2 (∃𝑦𝑥(𝜑𝑥𝑦) → ∃𝑤𝑥(𝑥𝑤𝜑))
2 biimpr 220 . . . 4 ((𝑥𝑤𝜑) → (𝜑𝑥𝑤))
32alimi 1807 . . 3 (∀𝑥(𝑥𝑤𝜑) → ∀𝑥(𝜑𝑥𝑤))
43eximi 1831 . 2 (∃𝑤𝑥(𝑥𝑤𝜑) → ∃𝑤𝑥(𝜑𝑥𝑤))
5 sepexlem 5304 . 2 (∃𝑤𝑥(𝜑𝑥𝑤) → ∃𝑧𝑥(𝑥𝑧𝜑))
61, 4, 53syl 18 1 (∃𝑦𝑥(𝜑𝑥𝑦) → ∃𝑧𝑥(𝑥𝑧𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wal 1534  wex 1775
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-sep 5301
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1776
This theorem is referenced by:  sepexi  5306
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