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Theorem 2eu2 2285
 Description: Double existential uniqueness. (Contributed by NM, 3-Dec-2001.)
Assertion
Ref Expression
2eu2 (∃!yxφ → (∃!x∃!yφ∃!xyφ))

Proof of Theorem 2eu2
StepHypRef Expression
1 eumo 2244 . . 3 (∃!yxφ∃*yxφ)
2 2moex 2275 . . 3 (∃*yxφx∃*yφ)
3 2eu1 2284 . . . 4 (x∃*yφ → (∃!x∃!yφ ↔ (∃!xyφ ∃!yxφ)))
4 simpl 443 . . . 4 ((∃!xyφ ∃!yxφ) → ∃!xyφ)
53, 4syl6bi 219 . . 3 (x∃*yφ → (∃!x∃!yφ∃!xyφ))
61, 2, 53syl 18 . 2 (∃!yxφ → (∃!x∃!yφ∃!xyφ))
7 2exeu 2281 . . 3 ((∃!xyφ ∃!yxφ) → ∃!x∃!yφ)
87expcom 424 . 2 (∃!yxφ → (∃!xyφ∃!x∃!yφ))
96, 8impbid 183 1 (∃!yxφ → (∃!x∃!yφ∃!xyφ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ 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:  2eu8  2291
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