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Theorem 2eu1 2582
Description: Double existential uniqueness. This theorem shows a condition under which a "naive" definition matches the correct one. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Wolf Lammen, 11-Nov-2019.)
Assertion
Ref Expression
2eu1 (∀𝑥∃*𝑦𝜑 → (∃!𝑥∃!𝑦𝜑 ↔ (∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑)))

Proof of Theorem 2eu1
StepHypRef Expression
1 2eu2ex 2575 . . . . 5 (∃!𝑥∃!𝑦𝜑 → ∃𝑥𝑦𝜑)
2 df-mo 2503 . . . . . . 7 (∃*𝑦𝜑 ↔ (∃𝑦𝜑 → ∃!𝑦𝜑))
32albii 1787 . . . . . 6 (∀𝑥∃*𝑦𝜑 ↔ ∀𝑥(∃𝑦𝜑 → ∃!𝑦𝜑))
4 euim 2552 . . . . . . 7 ((∃𝑥𝑦𝜑 ∧ ∀𝑥(∃𝑦𝜑 → ∃!𝑦𝜑)) → (∃!𝑥∃!𝑦𝜑 → ∃!𝑥𝑦𝜑))
54ex 449 . . . . . 6 (∃𝑥𝑦𝜑 → (∀𝑥(∃𝑦𝜑 → ∃!𝑦𝜑) → (∃!𝑥∃!𝑦𝜑 → ∃!𝑥𝑦𝜑)))
63, 5syl5bi 232 . . . . 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 2euswap 2577 . . . 4 (∀𝑥∃*𝑦𝜑 → (∃!𝑥𝑦𝜑 → ∃!𝑦𝑥𝜑))
108, 9syld 47 . . 3 (∀𝑥∃*𝑦𝜑 → (∃!𝑥∃!𝑦𝜑 → ∃!𝑦𝑥𝜑))
118, 10jcad 554 . 2 (∀𝑥∃*𝑦𝜑 → (∃!𝑥∃!𝑦𝜑 → (∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑)))
12 2exeu 2578 . 2 ((∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑) → ∃!𝑥∃!𝑦𝜑)
1311, 12impbid1 215 1 (∀𝑥∃*𝑦𝜑 → (∃!𝑥∃!𝑦𝜑 ↔ (∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  wal 1521  wex 1744  ∃!weu 2498  ∃*wmo 2499
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-eu 2502  df-mo 2503
This theorem is referenced by:  2eu2  2583  2eu3  2584  2eu5  2586
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