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Theorem 2moswap 2279
 Description: A condition allowing swap of "at most one" and existential quantifiers. (Contributed by NM, 10-Apr-2004.)
Assertion
Ref Expression
2moswap (x∃*yφ → (∃*xyφ∃*yxφ))

Proof of Theorem 2moswap
StepHypRef Expression
1 nfe1 1732 . . . 4 yyφ
21moexex 2273 . . 3 ((∃*xyφ x∃*yφ) → ∃*yx(yφ φ))
32expcom 424 . 2 (x∃*yφ → (∃*xyφ∃*yx(yφ φ)))
4 19.8a 1756 . . . . 5 (φyφ)
54pm4.71ri 614 . . . 4 (φ ↔ (yφ φ))
65exbii 1582 . . 3 (xφx(yφ φ))
76mobii 2240 . 2 (∃*yxφ∃*yx(yφ φ))
83, 7syl6ibr 218 1 (x∃*yφ → (∃*xyφ∃*yxφ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358  ∀wal 1540  ∃wex 1541  ∃*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:  2euswap  2280  2eu1  2284
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