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Theorem 2exeuv 2714
 Description: Version of 2exeu 2728 with 𝑥 and 𝑦 distinct, but not requiring ax-13 2385. (Contributed by Wolf Lammen, 2-Oct-2023.)
Assertion
Ref Expression
2exeuv ((∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑) → ∃!𝑥∃!𝑦𝜑)
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem 2exeuv
StepHypRef Expression
1 eumo 2658 . . . 4 (∃!𝑥𝑦𝜑 → ∃*𝑥𝑦𝜑)
2 euex 2657 . . . . 5 (∃!𝑦𝜑 → ∃𝑦𝜑)
32moimi 2620 . . . 4 (∃*𝑥𝑦𝜑 → ∃*𝑥∃!𝑦𝜑)
41, 3syl 17 . . 3 (∃!𝑥𝑦𝜑 → ∃*𝑥∃!𝑦𝜑)
5 2euexv 2713 . . 3 (∃!𝑦𝑥𝜑 → ∃𝑥∃!𝑦𝜑)
64, 5anim12ci 613 . 2 ((∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑) → (∃𝑥∃!𝑦𝜑 ∧ ∃*𝑥∃!𝑦𝜑))
7 df-eu 2649 . 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 2613  ∃!weu 2648 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 2137  ax-11 2152  ax-12 2167 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 2615  df-eu 2649 This theorem is referenced by:  2eu1v  2734
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