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Theorem 2exeu 1989
Description: Double existential uniqueness implies double uniqueness quantification. (Contributed by NM, 3-Dec-2001.)
Assertion
Ref Expression
2exeu ((∃!xyφ ∃!yxφ) → ∃!x∃!yφ)

Proof of Theorem 2exeu
StepHypRef Expression
1 excom 1551 . . . . 5 (yxφxyφ)
2 hbe1 1381 . . . . . . . 8 (xφxxφ)
32hbmo 1936 . . . . . . 7 (∃*yxφx∃*yxφ)
4319.41h 1572 . . . . . 6 (x(yφ ∃*yxφ) ↔ (xyφ ∃*yxφ))
5 19.8a 1479 . . . . . . . . 9 (φxφ)
65moimi 1962 . . . . . . . 8 (∃*yxφ∃*yφ)
76anim2i 324 . . . . . . 7 ((yφ ∃*yxφ) → (yφ ∃*yφ))
87eximi 1488 . . . . . 6 (x(yφ ∃*yxφ) → x(yφ ∃*yφ))
94, 8sylbir 125 . . . . 5 ((xyφ ∃*yxφ) → x(yφ ∃*yφ))
101, 9sylanb 268 . . . 4 ((yxφ ∃*yxφ) → x(yφ ∃*yφ))
11 simpl 102 . . . . . 6 ((yφ ∃*yφ) → yφ)
1211moimi 1962 . . . . 5 (∃*xyφ∃*x(yφ ∃*yφ))
1312adantl 262 . . . 4 ((xyφ ∃*xyφ) → ∃*x(yφ ∃*yφ))
1410, 13anim12i 321 . . 3 (((yxφ ∃*yxφ) (xyφ ∃*xyφ)) → (x(yφ ∃*yφ) ∃*x(yφ ∃*yφ)))
1514ancoms 255 . 2 (((xyφ ∃*xyφ) (yxφ ∃*yxφ)) → (x(yφ ∃*yφ) ∃*x(yφ ∃*yφ)))
16 eu5 1944 . . 3 (∃!xyφ ↔ (xyφ ∃*xyφ))
17 eu5 1944 . . 3 (∃!yxφ ↔ (yxφ ∃*yxφ))
1816, 17anbi12i 433 . 2 ((∃!xyφ ∃!yxφ) ↔ ((xyφ ∃*xyφ) (yxφ ∃*yxφ)))
19 eu5 1944 . . 3 (∃!x∃!yφ ↔ (x∃!yφ ∃*x∃!yφ))
20 eu5 1944 . . . . 5 (∃!yφ ↔ (yφ ∃*yφ))
2120exbii 1493 . . . 4 (x∃!yφx(yφ ∃*yφ))
2220mobii 1934 . . . 4 (∃*x∃!yφ∃*x(yφ ∃*yφ))
2321, 22anbi12i 433 . . 3 ((x∃!yφ ∃*x∃!yφ) ↔ (x(yφ ∃*yφ) ∃*x(yφ ∃*yφ)))
2419, 23bitri 173 . 2 (∃!x∃!yφ ↔ (x(yφ ∃*yφ) ∃*x(yφ ∃*yφ)))
2515, 18, 243imtr4i 190 1 ((∃!xyφ ∃!yxφ) → ∃!x∃!yφ)
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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