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Theorem 2exeu 2650
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 2373. Use the weaker 2exeuv 2636 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 2580 . . . 4 (∃!𝑥𝑦𝜑 → ∃*𝑥𝑦𝜑)
2 euex 2579 . . . . 5 (∃!𝑦𝜑 → ∃𝑦𝜑)
32moimi 2546 . . . 4 (∃*𝑥𝑦𝜑 → ∃*𝑥∃!𝑦𝜑)
41, 3syl 17 . . 3 (∃!𝑥𝑦𝜑 → ∃*𝑥∃!𝑦𝜑)
5 2euex 2645 . . 3 (∃!𝑦𝑥𝜑 → ∃𝑥∃!𝑦𝜑)
64, 5anim12ci 617 . 2 ((∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑) → (∃𝑥∃!𝑦𝜑 ∧ ∃*𝑥∃!𝑦𝜑))
7 df-eu 2571 . 2 (∃!𝑥∃!𝑦𝜑 ↔ (∃𝑥∃!𝑦𝜑 ∧ ∃*𝑥∃!𝑦𝜑))
86, 7sylibr 237 1 ((∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑) → ∃!𝑥∃!𝑦𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  wex 1786  ∃*wmo 2539  ∃!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 2020  ax-10 2145  ax-11 2162  ax-12 2179  ax-13 2373
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-tru 1545  df-ex 1787  df-nf 1791  df-mo 2541  df-eu 2571
This theorem is referenced by:  2eu1  2654  2eu2  2656  2eu3  2657
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