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Theorem 2euswapdc 1988
Description: A condition allowing swap of uniqueness and existential quantifiers. (Contributed by Jim Kingdon, 7-Jul-2018.)
Assertion
Ref Expression
2euswapdc (DECID xyφ → (x∃*yφ → (∃!xyφ∃!yxφ)))

Proof of Theorem 2euswapdc
StepHypRef Expression
1 excomim 1550 . . . . 5 (xyφyxφ)
21a1i 9 . . . 4 ((DECID xyφ x∃*yφ) → (xyφyxφ))
3 2moswapdc 1987 . . . . 5 (DECID xyφ → (x∃*yφ → (∃*xyφ∃*yxφ)))
43imp 115 . . . 4 ((DECID xyφ x∃*yφ) → (∃*xyφ∃*yxφ))
52, 4anim12d 318 . . 3 ((DECID xyφ x∃*yφ) → ((xyφ ∃*xyφ) → (yxφ ∃*yxφ)))
6 eu5 1944 . . 3 (∃!xyφ ↔ (xyφ ∃*xyφ))
7 eu5 1944 . . 3 (∃!yxφ ↔ (yxφ ∃*yxφ))
85, 6, 73imtr4g 194 . 2 ((DECID xyφ x∃*yφ) → (∃!xyφ∃!yxφ))
98ex 108 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  ∃!weu 1897  ∃*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-bndl 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:  euxfr2dc  2720  2reuswapdc  2737
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