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Theorem 2euex 1984
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 1944 . 2 (∃!xyφ ↔ (xyφ ∃*xyφ))
2 excom 1551 . . . 4 (xyφyxφ)
3 hbe1 1381 . . . . . 6 (yφyyφ)
43hbmo 1936 . . . . 5 (∃*xyφy∃*xyφ)
5 19.8a 1479 . . . . . . 7 (φyφ)
65moimi 1962 . . . . . 6 (∃*xyφ∃*xφ)
7 df-mo 1901 . . . . . 6 (∃*xφ ↔ (xφ∃!xφ))
86, 7sylib 127 . . . . 5 (∃*xyφ → (xφ∃!xφ))
94, 8eximdh 1499 . . . 4 (∃*xyφ → (yxφy∃!xφ))
102, 9syl5bi 141 . . 3 (∃*xyφ → (xyφy∃!xφ))
1110impcom 116 . 2 ((xyφ ∃*xyφ) → y∃!xφ)
121, 11sylbi 114 1 (∃!xyφy∃!xφ)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  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-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-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901
This theorem is referenced by: (None)
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