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Theorem eu6im 2638
 Description: One direction of eu6 2637 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 2008 . . 3 (∃𝑦𝑥(𝑥 = 𝑦𝜑) → ∃𝑥𝜑)
21anim1i 617 . 2 ((∃𝑦𝑥(𝑥 = 𝑦𝜑) ∧ ∃𝑧𝑥(𝜑𝑥 = 𝑧)) → (∃𝑥𝜑 ∧ ∃𝑧𝑥(𝜑𝑥 = 𝑧)))
3 eu6lem 2636 . 2 (∃𝑦𝑥(𝜑𝑥 = 𝑦) ↔ (∃𝑦𝑥(𝑥 = 𝑦𝜑) ∧ ∃𝑧𝑥(𝜑𝑥 = 𝑧)))
4 eu3v 2633 . 2 (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃𝑧𝑥(𝜑𝑥 = 𝑧)))
52, 3, 43imtr4i 295 1 (∃𝑦𝑥(𝜑𝑥 = 𝑦) → ∃!𝑥𝜑)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399  ∀wal 1536  ∃wex 1781  ∃!weu 2631 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-mo 2601  df-eu 2632 This theorem is referenced by: (None)
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