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Theorem 2eu1v 2651
Description: Double existential uniqueness. This theorem shows a condition under which a "naive" definition matches the correct one. Version of 2eu1 2650 with 𝑥 and 𝑦 distinct, but not requiring ax-13 2370. (Contributed by NM, 3-Dec-2001.) (Revised by Wolf Lammen, 2-Oct-2023.)
Assertion
Ref Expression
2eu1v (∀𝑥∃*𝑦𝜑 → (∃!𝑥∃!𝑦𝜑 ↔ (∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑)))
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem 2eu1v
StepHypRef Expression
1 2eu2ex 2643 . . . . 5 (∃!𝑥∃!𝑦𝜑 → ∃𝑥𝑦𝜑)
2 moeu 2581 . . . . . . . 8 (∃*𝑦𝜑 ↔ (∃𝑦𝜑 → ∃!𝑦𝜑))
32albii 1821 . . . . . . 7 (∀𝑥∃*𝑦𝜑 ↔ ∀𝑥(∃𝑦𝜑 → ∃!𝑦𝜑))
4 euim 2617 . . . . . . 7 ((∃𝑥𝑦𝜑 ∧ ∀𝑥(∃𝑦𝜑 → ∃!𝑦𝜑)) → (∃!𝑥∃!𝑦𝜑 → ∃!𝑥𝑦𝜑))
53, 4sylan2b 594 . . . . . 6 ((∃𝑥𝑦𝜑 ∧ ∀𝑥∃*𝑦𝜑) → (∃!𝑥∃!𝑦𝜑 → ∃!𝑥𝑦𝜑))
65ex 413 . . . . 5 (∃𝑥𝑦𝜑 → (∀𝑥∃*𝑦𝜑 → (∃!𝑥∃!𝑦𝜑 → ∃!𝑥𝑦𝜑)))
71, 6syl 17 . . . 4 (∃!𝑥∃!𝑦𝜑 → (∀𝑥∃*𝑦𝜑 → (∃!𝑥∃!𝑦𝜑 → ∃!𝑥𝑦𝜑)))
87pm2.43b 55 . . 3 (∀𝑥∃*𝑦𝜑 → (∃!𝑥∃!𝑦𝜑 → ∃!𝑥𝑦𝜑))
9 2euswapv 2630 . . . 4 (∀𝑥∃*𝑦𝜑 → (∃!𝑥𝑦𝜑 → ∃!𝑦𝑥𝜑))
108, 9syld 47 . . 3 (∀𝑥∃*𝑦𝜑 → (∃!𝑥∃!𝑦𝜑 → ∃!𝑦𝑥𝜑))
118, 10jcad 513 . 2 (∀𝑥∃*𝑦𝜑 → (∃!𝑥∃!𝑦𝜑 → (∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑)))
12 2exeuv 2632 . 2 ((∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑) → ∃!𝑥∃!𝑦𝜑)
1311, 12impbid1 224 1 (∀𝑥∃*𝑦𝜑 → (∃!𝑥∃!𝑦𝜑 ↔ (∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  wal 1539  wex 1781  ∃*wmo 2536  ∃!weu 2566
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-10 2137  ax-11 2154  ax-12 2171
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-ex 1782  df-nf 1786  df-mo 2538  df-eu 2567
This theorem is referenced by:  2eu5  2655
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