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Theorem euim 2552
Description: Add existential uniqueness 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 2551 . . 3 (∀𝑥(𝜑𝜓) → (∃!𝑥𝜓 → ∃*𝑥𝜑))
31, 2anim12ii 593 . 2 ((∃𝑥𝜑 ∧ ∀𝑥(𝜑𝜓)) → (∃!𝑥𝜓 → (∃𝑥𝜑 ∧ ∃*𝑥𝜑)))
4 eu5 2524 . 2 (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃*𝑥𝜑))
53, 4syl6ibr 242 1 ((∃𝑥𝜑 ∧ ∀𝑥(𝜑𝜓)) → (∃!𝑥𝜓 → ∃!𝑥𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  wal 1521  wex 1744  ∃!weu 2498  ∃*wmo 2499
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-10 2059  ax-12 2087
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-ex 1745  df-nf 1750  df-eu 2502  df-mo 2503
This theorem is referenced by:  2eu1  2582
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