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Theorem 2moswapdc 1987
Description: A condition allowing swap of "at most one" and existential quantifiers. (Contributed by Jim Kingdon, 6-Jul-2018.)
Assertion
Ref Expression
2moswapdc (DECID xyφ → (x∃*yφ → (∃*xyφ∃*yxφ)))

Proof of Theorem 2moswapdc
StepHypRef Expression
1 nfe1 1382 . . . 4 yyφ
21moexexdc 1981 . . 3 (DECID xyφ → ((∃*xyφ x∃*yφ) → ∃*yx(yφ φ)))
32expcomd 1327 . 2 (DECID xyφ → (x∃*yφ → (∃*xyφ∃*yx(yφ φ))))
4 19.8a 1479 . . . . . 6 (φyφ)
54pm4.71ri 372 . . . . 5 (φ ↔ (yφ φ))
65exbii 1493 . . . 4 (xφx(yφ φ))
76mobii 1934 . . 3 (∃*yxφ∃*yx(yφ φ))
87imbi2i 215 . 2 ((∃*xyφ∃*yxφ) ↔ (∃*xyφ∃*yx(yφ φ)))
93, 8syl6ibr 151 1 (DECID xyφ → (x∃*yφ → (∃*xyφ∃*yxφ)))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  DECID wdc 741  wal 1240  wex 1378  ∃*wmo 1898
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425
This theorem depends on definitions:  df-bi 110  df-dc 742  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901
This theorem is referenced by:  2euswapdc  1988
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