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Theorem 2euex 2276
 Description: Double quantification with existential uniqueness. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
Assertion
Ref Expression
2euex (∃!xyφy∃!xφ)

Proof of Theorem 2euex
StepHypRef Expression
1 eu5 2242 . 2 (∃!xyφ ↔ (xyφ ∃*xyφ))
2 excom 1741 . . . 4 (xyφyxφ)
3 nfe1 1732 . . . . . 6 yyφ
43nfmo 2221 . . . . 5 y∃*xyφ
5 19.8a 1756 . . . . . . 7 (φyφ)
65moimi 2251 . . . . . 6 (∃*xyφ∃*xφ)
7 df-mo 2209 . . . . . 6 (∃*xφ ↔ (xφ∃!xφ))
86, 7sylib 188 . . . . 5 (∃*xyφ → (xφ∃!xφ))
94, 8eximd 1770 . . . 4 (∃*xyφ → (yxφy∃!xφ))
102, 9syl5bi 208 . . 3 (∃*xyφ → (xyφy∃!xφ))
1110impcom 419 . 2 ((xyφ ∃*xyφ) → y∃!xφ)
121, 11sylbi 187 1 (∃!xyφy∃!xφ)
 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:  2exeu  2281
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