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Theorem 2exeu 2281
 Description: Double existential uniqueness implies double uniqueness quantification. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Mario Carneiro, 22-Dec-2016.)
Assertion
Ref Expression
2exeu ((∃!xyφ ∃!yxφ) → ∃!x∃!yφ)

Proof of Theorem 2exeu
StepHypRef Expression
1 eumo 2244 . . . 4 (∃!xyφ∃*xyφ)
2 euex 2227 . . . . 5 (∃!yφyφ)
32moimi 2251 . . . 4 (∃*xyφ∃*x∃!yφ)
41, 3syl 15 . . 3 (∃!xyφ∃*x∃!yφ)
5 2euex 2276 . . 3 (∃!yxφx∃!yφ)
64, 5anim12ci 550 . 2 ((∃!xyφ ∃!yxφ) → (x∃!yφ ∃*x∃!yφ))
7 eu5 2242 . 2 (∃!x∃!yφ ↔ (x∃!yφ ∃*x∃!yφ))
86, 7sylibr 203 1 ((∃!xyφ ∃!yxφ) → ∃!x∃!yφ)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358  ∃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:  2eu1  2284  2eu2  2285  2eu3  2286
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