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Theorem 2eu3 2286
 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 2215 . . . . 5 y∃*yφ
2119.31 1876 . . . 4 (y(∃*xφ ∃*yφ) ↔ (y∃*xφ ∃*yφ))
32albii 1566 . . 3 (xy(∃*xφ ∃*yφ) ↔ x(y∃*xφ ∃*yφ))
4 nfmo1 2215 . . . . 5 x∃*xφ
54nfal 1842 . . . 4 xy∃*xφ
6519.32 1875 . . 3 (x(y∃*xφ ∃*yφ) ↔ (y∃*xφ x∃*yφ))
73, 6bitri 240 . 2 (xy(∃*xφ ∃*yφ) ↔ (y∃*xφ x∃*yφ))
8 2eu1 2284 . . . . . . 7 (y∃*xφ → (∃!y∃!xφ ↔ (∃!yxφ ∃!xyφ)))
98biimpd 198 . . . . . 6 (y∃*xφ → (∃!y∃!xφ → (∃!yxφ ∃!xyφ)))
10 ancom 437 . . . . . 6 ((∃!yxφ ∃!xyφ) ↔ (∃!xyφ ∃!yxφ))
119, 10syl6ib 217 . . . . 5 (y∃*xφ → (∃!y∃!xφ → (∃!xyφ ∃!yxφ)))
1211adantld 453 . . . 4 (y∃*xφ → ((∃!x∃!yφ ∃!y∃!xφ) → (∃!xyφ ∃!yxφ)))
13 2eu1 2284 . . . . . 6 (x∃*yφ → (∃!x∃!yφ ↔ (∃!xyφ ∃!yxφ)))
1413biimpd 198 . . . . 5 (x∃*yφ → (∃!x∃!yφ → (∃!xyφ ∃!yxφ)))
1514adantrd 454 . . . 4 (x∃*yφ → ((∃!x∃!yφ ∃!y∃!xφ) → (∃!xyφ ∃!yxφ)))
1612, 15jaoi 368 . . 3 ((y∃*xφ x∃*yφ) → ((∃!x∃!yφ ∃!y∃!xφ) → (∃!xyφ ∃!yxφ)))
17 2exeu 2281 . . . 4 ((∃!xyφ ∃!yxφ) → ∃!x∃!yφ)
18 2exeu 2281 . . . . 5 ((∃!yxφ ∃!xyφ) → ∃!y∃!xφ)
1918ancoms 439 . . . 4 ((∃!xyφ ∃!yxφ) → ∃!y∃!xφ)
2017, 19jca 518 . . 3 ((∃!xyφ ∃!yxφ) → (∃!x∃!yφ ∃!y∃!xφ))
2116, 20impbid1 194 . 2 ((y∃*xφ x∃*yφ) → ((∃!x∃!yφ ∃!y∃!xφ) ↔ (∃!xyφ ∃!yxφ)))
227, 21sylbi 187 1 (xy(∃*xφ ∃*yφ) → ((∃!x∃!yφ ∃!y∃!xφ) ↔ (∃!xyφ ∃!yxφ)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∨ wo 357   ∧ wa 358  ∀wal 1540  ∃wex 1541  ∃!weu 2204  ∃*wmo 2205 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209 This theorem is referenced by: (None)
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