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Theorem exists1 2293
 Description: Two ways to express "only one thing exists." The left-hand side requires only one variable to express this. Both sides are false in set theory; see theorem dtru in set.mm. (Contributed by NM, 5-Apr-2004.)
Assertion
Ref Expression
exists1 (∃!x x = xx x = y)
Distinct variable group:   x,y

Proof of Theorem exists1
StepHypRef Expression
1 df-eu 2208 . 2 (∃!x x = xyx(x = xx = y))
2 equid 1676 . . . . . 6 x = x
32tbt 333 . . . . 5 (x = y ↔ (x = yx = x))
4 bicom 191 . . . . 5 ((x = yx = x) ↔ (x = xx = y))
53, 4bitri 240 . . . 4 (x = y ↔ (x = xx = y))
65albii 1566 . . 3 (x x = yx(x = xx = y))
76exbii 1582 . 2 (yx x = yyx(x = xx = y))
8 nfae 1954 . . 3 yx x = y
9819.9 1783 . 2 (yx x = yx x = y)
101, 7, 93bitr2i 264 1 (∃!x x = xx x = y)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176  ∀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-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-eu 2208 This theorem is referenced by:  exists2  2294
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