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Theorem 2eu3 2584
Description: Double existential uniqueness. (Contributed by NM, 3-Dec-2001.)
Assertion
Ref Expression
2eu3 (∀𝑥𝑦(∃*𝑥𝜑 ∨ ∃*𝑦𝜑) → ((∃!𝑥∃!𝑦𝜑 ∧ ∃!𝑦∃!𝑥𝜑) ↔ (∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑)))

Proof of Theorem 2eu3
StepHypRef Expression
1 nfmo1 2509 . . . . 5 𝑦∃*𝑦𝜑
2119.31 2140 . . . 4 (∀𝑦(∃*𝑥𝜑 ∨ ∃*𝑦𝜑) ↔ (∀𝑦∃*𝑥𝜑 ∨ ∃*𝑦𝜑))
32albii 1787 . . 3 (∀𝑥𝑦(∃*𝑥𝜑 ∨ ∃*𝑦𝜑) ↔ ∀𝑥(∀𝑦∃*𝑥𝜑 ∨ ∃*𝑦𝜑))
4 nfmo1 2509 . . . . 5 𝑥∃*𝑥𝜑
54nfal 2191 . . . 4 𝑥𝑦∃*𝑥𝜑
6519.32 2139 . . 3 (∀𝑥(∀𝑦∃*𝑥𝜑 ∨ ∃*𝑦𝜑) ↔ (∀𝑦∃*𝑥𝜑 ∨ ∀𝑥∃*𝑦𝜑))
73, 6bitri 264 . 2 (∀𝑥𝑦(∃*𝑥𝜑 ∨ ∃*𝑦𝜑) ↔ (∀𝑦∃*𝑥𝜑 ∨ ∀𝑥∃*𝑦𝜑))
8 2eu1 2582 . . . . . . 7 (∀𝑦∃*𝑥𝜑 → (∃!𝑦∃!𝑥𝜑 ↔ (∃!𝑦𝑥𝜑 ∧ ∃!𝑥𝑦𝜑)))
98biimpd 219 . . . . . 6 (∀𝑦∃*𝑥𝜑 → (∃!𝑦∃!𝑥𝜑 → (∃!𝑦𝑥𝜑 ∧ ∃!𝑥𝑦𝜑)))
10 ancom 465 . . . . . 6 ((∃!𝑦𝑥𝜑 ∧ ∃!𝑥𝑦𝜑) ↔ (∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑))
119, 10syl6ib 241 . . . . 5 (∀𝑦∃*𝑥𝜑 → (∃!𝑦∃!𝑥𝜑 → (∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑)))
1211adantld 482 . . . 4 (∀𝑦∃*𝑥𝜑 → ((∃!𝑥∃!𝑦𝜑 ∧ ∃!𝑦∃!𝑥𝜑) → (∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑)))
13 2eu1 2582 . . . . . 6 (∀𝑥∃*𝑦𝜑 → (∃!𝑥∃!𝑦𝜑 ↔ (∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑)))
1413biimpd 219 . . . . 5 (∀𝑥∃*𝑦𝜑 → (∃!𝑥∃!𝑦𝜑 → (∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑)))
1514adantrd 483 . . . 4 (∀𝑥∃*𝑦𝜑 → ((∃!𝑥∃!𝑦𝜑 ∧ ∃!𝑦∃!𝑥𝜑) → (∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑)))
1612, 15jaoi 393 . . 3 ((∀𝑦∃*𝑥𝜑 ∨ ∀𝑥∃*𝑦𝜑) → ((∃!𝑥∃!𝑦𝜑 ∧ ∃!𝑦∃!𝑥𝜑) → (∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑)))
17 2exeu 2578 . . . 4 ((∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑) → ∃!𝑥∃!𝑦𝜑)
18 2exeu 2578 . . . . 5 ((∃!𝑦𝑥𝜑 ∧ ∃!𝑥𝑦𝜑) → ∃!𝑦∃!𝑥𝜑)
1918ancoms 468 . . . 4 ((∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑) → ∃!𝑦∃!𝑥𝜑)
2017, 19jca 553 . . 3 ((∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑) → (∃!𝑥∃!𝑦𝜑 ∧ ∃!𝑦∃!𝑥𝜑))
2116, 20impbid1 215 . 2 ((∀𝑦∃*𝑥𝜑 ∨ ∀𝑥∃*𝑦𝜑) → ((∃!𝑥∃!𝑦𝜑 ∧ ∃!𝑦∃!𝑥𝜑) ↔ (∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑)))
227, 21sylbi 207 1 (∀𝑥𝑦(∃*𝑥𝜑 ∨ ∃*𝑦𝜑) → ((∃!𝑥∃!𝑦𝜑 ∧ ∃!𝑦∃!𝑥𝜑) ↔ (∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wo 382  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: (None)
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