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Theorem 2euswap 2280
 Description: A condition allowing swap of uniqueness and existential quantifiers. (Contributed by NM, 10-Apr-2004.)
Assertion
Ref Expression
2euswap (x∃*yφ → (∃!xyφ∃!yxφ))

Proof of Theorem 2euswap
StepHypRef Expression
1 excomim 1742 . . . 4 (xyφyxφ)
21a1i 10 . . 3 (x∃*yφ → (xyφyxφ))
3 2moswap 2279 . . 3 (x∃*yφ → (∃*xyφ∃*yxφ))
42, 3anim12d 546 . 2 (x∃*yφ → ((xyφ ∃*xyφ) → (yxφ ∃*yxφ)))
5 eu5 2242 . 2 (∃!xyφ ↔ (xyφ ∃*xyφ))
6 eu5 2242 . 2 (∃!yxφ ↔ (yxφ ∃*yxφ))
74, 5, 63imtr4g 261 1 (x∃*yφ → (∃!xyφ∃!yxφ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358  ∀wal 1540  ∃wex 1541  ∃!weu 2204  ∃*wmo 2205 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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:  euxfr2  3021  2reuswap  3038
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