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Theorem 2eu4 1966
Description: This theorem provides us with a definition of double existential uniqueness ("exactly one x and exactly one y"). Naively one might think (incorrectly) that it could be defined by ∃!x∃!yφ. See 2exeu 1965 for a one-way implication. (Contributed by NM, 3-Dec-2001.)
Assertion
Ref Expression
2eu4 ((∃!xyφ ∃!yxφ) ↔ (xyφ zwxy(φ → (x = z y = w))))
Distinct variable groups:   x,y,z,w   φ,z,w
Allowed substitution hints:   φ(x,y)

Proof of Theorem 2eu4
StepHypRef Expression
1 ax-17 1392 . . . 4 (yφzyφ)
21eu3h 1918 . . 3 (∃!xyφ ↔ (xyφ zx(yφx = z)))
3 ax-17 1392 . . . 4 (xφwxφ)
43eu3h 1918 . . 3 (∃!yxφ ↔ (yxφ wy(xφy = w)))
52, 4anbi12i 433 . 2 ((∃!xyφ ∃!yxφ) ↔ ((xyφ zx(yφx = z)) (yxφ wy(xφy = w))))
6 an4 505 . 2 (((xyφ zx(yφx = z)) (yxφ wy(xφy = w))) ↔ ((xyφ yxφ) (zx(yφx = z) wy(xφy = w))))
7 excom 1527 . . . . 5 (yxφxyφ)
87anbi2i 430 . . . 4 ((xyφ yxφ) ↔ (xyφ xyφ))
9 anidm 376 . . . 4 ((xyφ xyφ) ↔ xyφ)
108, 9bitri 173 . . 3 ((xyφ yxφ) ↔ xyφ)
11 hba1 1406 . . . . . . . . . 10 (xy(φy = w) → xxy(φy = w))
121119.3h 1418 . . . . . . . . 9 (xxy(φy = w) ↔ xy(φy = w))
1312anbi2i 430 . . . . . . . 8 ((xy(φx = z) xxy(φy = w)) ↔ (xy(φx = z) xy(φy = w)))
14 19.26 1343 . . . . . . . 8 (x(y(φx = z) xy(φy = w)) ↔ (xy(φx = z) xxy(φy = w)))
15 jcab 520 . . . . . . . . . . . 12 ((φ → (x = z y = w)) ↔ ((φx = z) (φy = w)))
1615albii 1332 . . . . . . . . . . 11 (y(φ → (x = z y = w)) ↔ y((φx = z) (φy = w)))
17 19.26 1343 . . . . . . . . . . 11 (y((φx = z) (φy = w)) ↔ (y(φx = z) y(φy = w)))
1816, 17bitri 173 . . . . . . . . . 10 (y(φ → (x = z y = w)) ↔ (y(φx = z) y(φy = w)))
1918albii 1332 . . . . . . . . 9 (xy(φ → (x = z y = w)) ↔ x(y(φx = z) y(φy = w)))
20 19.26 1343 . . . . . . . . 9 (x(y(φx = z) y(φy = w)) ↔ (xy(φx = z) xy(φy = w)))
2119, 20bitri 173 . . . . . . . 8 (xy(φ → (x = z y = w)) ↔ (xy(φx = z) xy(φy = w)))
2213, 14, 213bitr4ri 202 . . . . . . 7 (xy(φ → (x = z y = w)) ↔ x(y(φx = z) xy(φy = w)))
23 19.26 1343 . . . . . . . . 9 (y(y(φx = z) x(φy = w)) ↔ (yy(φx = z) yx(φy = w)))
24 hba1 1406 . . . . . . . . . . 11 (y(φx = z) → yy(φx = z))
252419.3h 1418 . . . . . . . . . 10 (yy(φx = z) ↔ y(φx = z))
26 alcom 1340 . . . . . . . . . 10 (yx(φy = w) ↔ xy(φy = w))
2725, 26anbi12i 433 . . . . . . . . 9 ((yy(φx = z) yx(φy = w)) ↔ (y(φx = z) xy(φy = w)))
2823, 27bitri 173 . . . . . . . 8 (y(y(φx = z) x(φy = w)) ↔ (y(φx = z) xy(φy = w)))
2928albii 1332 . . . . . . 7 (xy(y(φx = z) x(φy = w)) ↔ x(y(φx = z) xy(φy = w)))
3022, 29bitr4i 176 . . . . . 6 (xy(φ → (x = z y = w)) ↔ xy(y(φx = z) x(φy = w)))
31 19.23v 1736 . . . . . . . 8 (y(φx = z) ↔ (yφx = z))
32 19.23v 1736 . . . . . . . 8 (x(φy = w) ↔ (xφy = w))
3331, 32anbi12i 433 . . . . . . 7 ((y(φx = z) x(φy = w)) ↔ ((yφx = z) (xφy = w)))
34332albii 1333 . . . . . 6 (xy(y(φx = z) x(φy = w)) ↔ xy((yφx = z) (xφy = w)))
35 hbe1 1357 . . . . . . . 8 (yφyyφ)
36 ax-17 1392 . . . . . . . 8 (x = zy x = z)
3735, 36hbim 1410 . . . . . . 7 ((yφx = z) → y(yφx = z))
38 hbe1 1357 . . . . . . . 8 (xφxxφ)
39 ax-17 1392 . . . . . . . 8 (y = wx y = w)
4038, 39hbim 1410 . . . . . . 7 ((xφy = w) → x(xφy = w))
4137, 40aaanh 1451 . . . . . 6 (xy((yφx = z) (xφy = w)) ↔ (x(yφx = z) y(xφy = w)))
4230, 34, 413bitri 195 . . . . 5 (xy(φ → (x = z y = w)) ↔ (x(yφx = z) y(xφy = w)))
43422exbii 1470 . . . 4 (zwxy(φ → (x = z y = w)) ↔ zw(x(yφx = z) y(xφy = w)))
44 eeanv 1780 . . . 4 (zw(x(yφx = z) y(xφy = w)) ↔ (zx(yφx = z) wy(xφy = w)))
4543, 44bitr2i 174 . . 3 ((zx(yφx = z) wy(xφy = w)) ↔ zwxy(φ → (x = z y = w)))
4610, 45anbi12i 433 . 2 (((xyφ yxφ) (zx(yφx = z) wy(xφy = w))) ↔ (xyφ zwxy(φ → (x = z y = w))))
475, 6, 463bitri 195 1 ((∃!xyφ ∃!yxφ) ↔ (xyφ zwxy(φ → (x = z y = w))))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98  wal 1221  wex 1354  ∃!weu 1873
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 614  ax-5 1309  ax-7 1310  ax-gen 1311  ax-ie1 1355  ax-ie2 1356  ax-8 1368  ax-10 1369  ax-11 1370  ax-i12 1371  ax-bnd 1372  ax-4 1373  ax-17 1392  ax-i9 1396  ax-ial 1400  ax-i5r 1401
This theorem depends on definitions:  df-bi 110  df-nf 1323  df-sb 1619  df-eu 1876
This theorem is referenced by: (None)
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