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Theorem euim 2705
 Description: Add unique existential quantifiers to an implication. Note the reversed implication in the antecedent. (Contributed by NM, 19-Oct-2005.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
Assertion
Ref Expression
euim ((∃𝑥𝜑 ∧ ∀𝑥(𝜑𝜓)) → (∃!𝑥𝜓 → ∃!𝑥𝜑))

Proof of Theorem euim
StepHypRef Expression
1 ax-1 6 . . 3 (∃𝑥𝜑 → (∃!𝑥𝜓 → ∃𝑥𝜑))
2 euimmo 2704 . . 3 (∀𝑥(𝜑𝜓) → (∃!𝑥𝜓 → ∃*𝑥𝜑))
31, 2anim12ii 613 . 2 ((∃𝑥𝜑 ∧ ∀𝑥(𝜑𝜓)) → (∃!𝑥𝜓 → (∃𝑥𝜑 ∧ ∃*𝑥𝜑)))
4 df-eu 2640 . 2 (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃*𝑥𝜑))
53, 4syl6ibr 244 1 ((∃𝑥𝜑 ∧ ∀𝑥(𝜑𝜓)) → (∃!𝑥𝜓 → ∃!𝑥𝜑))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 386  ∀wal 1656  ∃wex 1880  ∃*wmo 2603  ∃!weu 2639 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011 This theorem depends on definitions:  df-bi 199  df-an 387  df-ex 1881  df-mo 2605  df-eu 2640 This theorem is referenced by:  2eu1  2733  dfatcolem  42157
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