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Theorem eu6im 2577
Description: One direction of eu6 2576 needs fewer axioms. (Contributed by Wolf Lammen, 2-Mar-2023.)
Assertion
Ref Expression
eu6im (∃𝑦𝑥(𝜑𝑥 = 𝑦) → ∃!𝑥𝜑)
Distinct variable groups:   𝑥,𝑦   𝜑,𝑦
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem eu6im
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 exsbim 2009 . . 3 (∃𝑦𝑥(𝑥 = 𝑦𝜑) → ∃𝑥𝜑)
21anim1i 615 . 2 ((∃𝑦𝑥(𝑥 = 𝑦𝜑) ∧ ∃𝑧𝑥(𝜑𝑥 = 𝑧)) → (∃𝑥𝜑 ∧ ∃𝑧𝑥(𝜑𝑥 = 𝑧)))
3 eu6lem 2575 . 2 (∃𝑦𝑥(𝜑𝑥 = 𝑦) ↔ (∃𝑦𝑥(𝑥 = 𝑦𝜑) ∧ ∃𝑧𝑥(𝜑𝑥 = 𝑧)))
4 eu3v 2572 . 2 (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃𝑧𝑥(𝜑𝑥 = 𝑧)))
52, 3, 43imtr4i 292 1 (∃𝑦𝑥(𝜑𝑥 = 𝑦) → ∃!𝑥𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  wal 1540  wex 1786  ∃!weu 2570
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015
This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1787  df-mo 2542  df-eu 2571
This theorem is referenced by:  eufsnlem  46135
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