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Theorem 2eu4 2287
 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 2eu1 2284 for a condition under which the naive definition holds and 2exeu 2281 for a one-way implication. See 2eu5 2288 and 2eu8 2291 for alternate definitions. (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 nfv 1619 . . . 4 zyφ
21eu3 2230 . . 3 (∃!xyφ ↔ (xyφ zx(yφx = z)))
3 nfv 1619 . . . 4 wxφ
43eu3 2230 . . 3 (∃!yxφ ↔ (yxφ wy(xφy = w)))
52, 4anbi12i 678 . 2 ((∃!xyφ ∃!yxφ) ↔ ((xyφ zx(yφx = z)) (yxφ wy(xφy = w))))
6 an4 797 . 2 (((xyφ zx(yφx = z)) (yxφ wy(xφy = w))) ↔ ((xyφ yxφ) (zx(yφx = z) wy(xφy = w))))
7 excom 1741 . . . . 5 (yxφxyφ)
87anbi2i 675 . . . 4 ((xyφ yxφ) ↔ (xyφ xyφ))
9 anidm 625 . . . 4 ((xyφ xyφ) ↔ xyφ)
108, 9bitri 240 . . 3 ((xyφ yxφ) ↔ xyφ)
11 19.26 1593 . . . . . . . 8 (x(y(φx = z) xy(φy = w)) ↔ (xy(φx = z) xxy(φy = w)))
12 nfa1 1788 . . . . . . . . . . 11 xxy(φy = w)
131219.3 1785 . . . . . . . . . 10 (xxy(φy = w) ↔ xy(φy = w))
1413anbi2i 675 . . . . . . . . 9 ((xy(φx = z) xxy(φy = w)) ↔ (xy(φx = z) xy(φy = w)))
15 jcab 833 . . . . . . . . . . . . 13 ((φ → (x = z y = w)) ↔ ((φx = z) (φy = w)))
1615albii 1566 . . . . . . . . . . . 12 (y(φ → (x = z y = w)) ↔ y((φx = z) (φy = w)))
17 19.26 1593 . . . . . . . . . . . 12 (y((φx = z) (φy = w)) ↔ (y(φx = z) y(φy = w)))
1816, 17bitri 240 . . . . . . . . . . 11 (y(φ → (x = z y = w)) ↔ (y(φx = z) y(φy = w)))
1918albii 1566 . . . . . . . . . 10 (xy(φ → (x = z y = w)) ↔ x(y(φx = z) y(φy = w)))
20 19.26 1593 . . . . . . . . . 10 (x(y(φx = z) y(φy = w)) ↔ (xy(φx = z) xy(φy = w)))
2119, 20bitri 240 . . . . . . . . 9 (xy(φ → (x = z y = w)) ↔ (xy(φx = z) xy(φy = w)))
2214, 21bitr4i 243 . . . . . . . 8 ((xy(φx = z) xxy(φy = w)) ↔ xy(φ → (x = z y = w)))
2311, 22bitr2i 241 . . . . . . 7 (xy(φ → (x = z y = w)) ↔ x(y(φx = z) xy(φy = w)))
24 19.26 1593 . . . . . . . . 9 (y(y(φx = z) x(φy = w)) ↔ (yy(φx = z) yx(φy = w)))
25 nfa1 1788 . . . . . . . . . . 11 yy(φx = z)
262519.3 1785 . . . . . . . . . 10 (yy(φx = z) ↔ y(φx = z))
27 alcom 1737 . . . . . . . . . 10 (yx(φy = w) ↔ xy(φy = w))
2826, 27anbi12i 678 . . . . . . . . 9 ((yy(φx = z) yx(φy = w)) ↔ (y(φx = z) xy(φy = w)))
2924, 28bitri 240 . . . . . . . 8 (y(y(φx = z) x(φy = w)) ↔ (y(φx = z) xy(φy = w)))
3029albii 1566 . . . . . . 7 (xy(y(φx = z) x(φy = w)) ↔ x(y(φx = z) xy(φy = w)))
3123, 30bitr4i 243 . . . . . 6 (xy(φ → (x = z y = w)) ↔ xy(y(φx = z) x(φy = w)))
32 19.23v 1891 . . . . . . . 8 (y(φx = z) ↔ (yφx = z))
33 19.23v 1891 . . . . . . . 8 (x(φy = w) ↔ (xφy = w))
3432, 33anbi12i 678 . . . . . . 7 ((y(φx = z) x(φy = w)) ↔ ((yφx = z) (xφy = w)))
35342albii 1567 . . . . . 6 (xy(y(φx = z) x(φy = w)) ↔ xy((yφx = z) (xφy = w)))
36 nfe1 1732 . . . . . . . 8 yyφ
37 nfv 1619 . . . . . . . 8 y x = z
3836, 37nfim 1813 . . . . . . 7 y(yφx = z)
39 nfe1 1732 . . . . . . . 8 xxφ
40 nfv 1619 . . . . . . . 8 x y = w
4139, 40nfim 1813 . . . . . . 7 x(xφy = w)
4238, 41aaan 1884 . . . . . 6 (xy((yφx = z) (xφy = w)) ↔ (x(yφx = z) y(xφy = w)))
4331, 35, 423bitri 262 . . . . 5 (xy(φ → (x = z y = w)) ↔ (x(yφx = z) y(xφy = w)))
44432exbii 1583 . . . 4 (zwxy(φ → (x = z y = w)) ↔ zw(x(yφx = z) y(xφy = w)))
45 eeanv 1913 . . . 4 (zw(x(yφx = z) y(xφy = w)) ↔ (zx(yφx = z) wy(xφy = w)))
4644, 45bitr2i 241 . . 3 ((zx(yφx = z) wy(xφy = w)) ↔ zwxy(φ → (x = z y = w)))
4710, 46anbi12i 678 . 2 (((xyφ yxφ) (zx(yφx = z) wy(xφy = w))) ↔ (xyφ zwxy(φ → (x = z y = w))))
485, 6, 473bitri 262 1 ((∃!xyφ ∃!yxφ) ↔ (xyφ zwxy(φ → (x = z y = w))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540  ∃wex 1541   = wceq 1642  ∃!weu 2204 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208 This theorem is referenced by:  2eu5  2288  2eu6  2289
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