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Theorem sepexlem 5304
Description: Lemma for sepex 5305. Use sepex 5305 instead. (Contributed by Matthew House, 19-Sep-2025.) (New usage is discouraged.)
Assertion
Ref Expression
sepexlem (∃𝑦𝑥(𝜑𝑥𝑦) → ∃𝑧𝑥(𝑥𝑧𝜑))
Distinct variable groups:   𝑥,𝑦,𝑧   𝜑,𝑦,𝑧
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem sepexlem
StepHypRef Expression
1 ax-sep 5301 . . 3 𝑧𝑥(𝑥𝑧 ↔ (𝑥𝑦𝜑))
2 bimsc1 844 . . . . . 6 (((𝜑𝑥𝑦) ∧ (𝑥𝑧 ↔ (𝑥𝑦𝜑))) → (𝑥𝑧𝜑))
32ex 412 . . . . 5 ((𝜑𝑥𝑦) → ((𝑥𝑧 ↔ (𝑥𝑦𝜑)) → (𝑥𝑧𝜑)))
43al2imi 1811 . . . 4 (∀𝑥(𝜑𝑥𝑦) → (∀𝑥(𝑥𝑧 ↔ (𝑥𝑦𝜑)) → ∀𝑥(𝑥𝑧𝜑)))
54eximdv 1914 . . 3 (∀𝑥(𝜑𝑥𝑦) → (∃𝑧𝑥(𝑥𝑧 ↔ (𝑥𝑦𝜑)) → ∃𝑧𝑥(𝑥𝑧𝜑)))
61, 5mpi 20 . 2 (∀𝑥(𝜑𝑥𝑦) → ∃𝑧𝑥(𝑥𝑧𝜑))
76exlimiv 1927 1 (∃𝑦𝑥(𝜑𝑥𝑦) → ∃𝑧𝑥(𝑥𝑧𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  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:  sepex  5305
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