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Theorem 2exeu 2637
Description: Double existential uniqueness implies double unique existential quantification. The converse does not hold. Usage of this theorem is discouraged because it depends on ax-13 2366. Use the weaker 2exeuv 2623 when possible. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Mario Carneiro, 22-Dec-2016.) (New usage is discouraged.)
Assertion
Ref Expression
2exeu ((∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑) → ∃!𝑥∃!𝑦𝜑)

Proof of Theorem 2exeu
StepHypRef Expression
1 eumo 2567 . . . 4 (∃!𝑥𝑦𝜑 → ∃*𝑥𝑦𝜑)
2 euex 2566 . . . . 5 (∃!𝑦𝜑 → ∃𝑦𝜑)
32moimi 2534 . . . 4 (∃*𝑥𝑦𝜑 → ∃*𝑥∃!𝑦𝜑)
41, 3syl 17 . . 3 (∃!𝑥𝑦𝜑 → ∃*𝑥∃!𝑦𝜑)
5 2euex 2632 . . 3 (∃!𝑦𝑥𝜑 → ∃𝑥∃!𝑦𝜑)
64, 5anim12ci 613 . 2 ((∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑) → (∃𝑥∃!𝑦𝜑 ∧ ∃*𝑥∃!𝑦𝜑))
7 df-eu 2558 . 2 (∃!𝑥∃!𝑦𝜑 ↔ (∃𝑥∃!𝑦𝜑 ∧ ∃*𝑥∃!𝑦𝜑))
86, 7sylibr 233 1 ((∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑) → ∃!𝑥∃!𝑦𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wex 1774  ∃*wmo 2527  ∃!weu 2557
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-10 2130  ax-11 2147  ax-12 2164  ax-13 2366
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-tru 1537  df-ex 1775  df-nf 1779  df-mo 2529  df-eu 2558
This theorem is referenced by:  2eu1  2641  2eu2  2643  2eu3  2644
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