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Theorem 2exeu 2730
Description: Double existential uniqueness implies double unique existential quantification. The converse does not hold. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Mario Carneiro, 22-Dec-2016.)
Assertion
Ref Expression
2exeu ((∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑) → ∃!𝑥∃!𝑦𝜑)

Proof of Theorem 2exeu
StepHypRef Expression
1 eumo 2661 . . . 4 (∃!𝑥𝑦𝜑 → ∃*𝑥𝑦𝜑)
2 euex 2660 . . . . 5 (∃!𝑦𝜑 → ∃𝑦𝜑)
32moimi 2625 . . . 4 (∃*𝑥𝑦𝜑 → ∃*𝑥∃!𝑦𝜑)
41, 3syl 17 . . 3 (∃!𝑥𝑦𝜑 → ∃*𝑥∃!𝑦𝜑)
5 2euex 2725 . . 3 (∃!𝑦𝑥𝜑 → ∃𝑥∃!𝑦𝜑)
64, 5anim12ci 613 . 2 ((∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑) → (∃𝑥∃!𝑦𝜑 ∧ ∃*𝑥∃!𝑦𝜑))
7 df-eu 2652 . 2 (∃!𝑥∃!𝑦𝜑 ↔ (∃𝑥∃!𝑦𝜑 ∧ ∃*𝑥∃!𝑦𝜑))
86, 7sylibr 235 1 ((∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑) → ∃!𝑥∃!𝑦𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wex 1773  ∃*wmo 2618  ∃!weu 2651
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-10 2138  ax-11 2153  ax-12 2169  ax-13 2385
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-tru 1533  df-ex 1774  df-nf 1778  df-mo 2620  df-eu 2652
This theorem is referenced by:  2eu1  2734  2eu1OLD  2735  2eu2  2737  2eu3  2738  2eu3OLD  2739
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