New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  2eu1 GIF version

Theorem 2eu1 2284
 Description: Double existential uniqueness. This theorem shows a condition under which a "naive" definition matches the correct one. (Contributed by NM, 3-Dec-2001.)
Assertion
Ref Expression
2eu1 (x∃*yφ → (∃!x∃!yφ ↔ (∃!xyφ ∃!yxφ)))

Proof of Theorem 2eu1
StepHypRef Expression
1 eu5 2242 . . . . . . . 8 (∃!x∃!yφ ↔ (x∃!yφ ∃*x∃!yφ))
2 eu5 2242 . . . . . . . . . 10 (∃!yφ ↔ (yφ ∃*yφ))
32exbii 1582 . . . . . . . . 9 (x∃!yφx(yφ ∃*yφ))
42mobii 2240 . . . . . . . . 9 (∃*x∃!yφ∃*x(yφ ∃*yφ))
53, 4anbi12i 678 . . . . . . . 8 ((x∃!yφ ∃*x∃!yφ) ↔ (x(yφ ∃*yφ) ∃*x(yφ ∃*yφ)))
61, 5bitri 240 . . . . . . 7 (∃!x∃!yφ ↔ (x(yφ ∃*yφ) ∃*x(yφ ∃*yφ)))
76simprbi 450 . . . . . 6 (∃!x∃!yφ∃*x(yφ ∃*yφ))
8 sp 1747 . . . . . . . . . . . 12 (x∃*yφ∃*yφ)
98anim2i 552 . . . . . . . . . . 11 ((yφ x∃*yφ) → (yφ ∃*yφ))
109ancoms 439 . . . . . . . . . 10 ((x∃*yφ yφ) → (yφ ∃*yφ))
1110moimi 2251 . . . . . . . . 9 (∃*x(yφ ∃*yφ) → ∃*x(x∃*yφ yφ))
12 nfa1 1788 . . . . . . . . . 10 xx∃*yφ
1312moanim 2260 . . . . . . . . 9 (∃*x(x∃*yφ yφ) ↔ (x∃*yφ∃*xyφ))
1411, 13sylib 188 . . . . . . . 8 (∃*x(yφ ∃*yφ) → (x∃*yφ∃*xyφ))
1514ancrd 537 . . . . . . 7 (∃*x(yφ ∃*yφ) → (x∃*yφ → (∃*xyφ x∃*yφ)))
16 2moswap 2279 . . . . . . . . 9 (x∃*yφ → (∃*xyφ∃*yxφ))
1716com12 27 . . . . . . . 8 (∃*xyφ → (x∃*yφ∃*yxφ))
1817imdistani 671 . . . . . . 7 ((∃*xyφ x∃*yφ) → (∃*xyφ ∃*yxφ))
1915, 18syl6 29 . . . . . 6 (∃*x(yφ ∃*yφ) → (x∃*yφ → (∃*xyφ ∃*yxφ)))
207, 19syl 15 . . . . 5 (∃!x∃!yφ → (x∃*yφ → (∃*xyφ ∃*yxφ)))
21 2eu2ex 2278 . . . . . 6 (∃!x∃!yφxyφ)
22 excom 1741 . . . . . . 7 (xyφyxφ)
2321, 22sylib 188 . . . . . 6 (∃!x∃!yφyxφ)
2421, 23jca 518 . . . . 5 (∃!x∃!yφ → (xyφ yxφ))
2520, 24jctild 527 . . . 4 (∃!x∃!yφ → (x∃*yφ → ((xyφ yxφ) (∃*xyφ ∃*yxφ))))
26 eu5 2242 . . . . . 6 (∃!xyφ ↔ (xyφ ∃*xyφ))
27 eu5 2242 . . . . . 6 (∃!yxφ ↔ (yxφ ∃*yxφ))
2826, 27anbi12i 678 . . . . 5 ((∃!xyφ ∃!yxφ) ↔ ((xyφ ∃*xyφ) (yxφ ∃*yxφ)))
29 an4 797 . . . . 5 (((xyφ ∃*xyφ) (yxφ ∃*yxφ)) ↔ ((xyφ yxφ) (∃*xyφ ∃*yxφ)))
3028, 29bitri 240 . . . 4 ((∃!xyφ ∃!yxφ) ↔ ((xyφ yxφ) (∃*xyφ ∃*yxφ)))
3125, 30syl6ibr 218 . . 3 (∃!x∃!yφ → (x∃*yφ → (∃!xyφ ∃!yxφ)))
3231com12 27 . 2 (x∃*yφ → (∃!x∃!yφ → (∃!xyφ ∃!yxφ)))
33 2exeu 2281 . 2 ((∃!xyφ ∃!yxφ) → ∃!x∃!yφ)
3432, 33impbid1 194 1 (x∃*yφ → (∃!x∃!yφ ↔ (∃!xyφ ∃!yxφ)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540  ∃wex 1541  ∃!weu 2204  ∃*wmo 2205 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  df-mo 2209 This theorem is referenced by:  2eu2  2285  2eu3  2286  2eu5  2288
 Copyright terms: Public domain W3C validator