HomeHome Metamath Proof Explorer < Previous   Next >
Related theorems
Unicode version
Theorem exists1 1455
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 2767.
Assertion
Ref Expression
exists1 |- (E!x x = x <-> A.x x = y)
Distinct variable group:   x,y
Proof of Theorem exists1
StepHypRef Expression
1 df-eu 1380 . 2 |- (E!x x = x <-> E.yA.x(x = x <-> x = y))
2 equid 1124 . . . . . 6 |- x = x
32tbt 719 . . . . 5 |- (x = y <-> (x = y <-> x = x))
4 bicom 519 . . . . 5 |- ((x = y <-> x = x) <-> (x = x <-> x = y))
53, 4bitr 173 . . . 4 |- (x = y <-> (x = x <-> x = y))
65albii 997 . . 3 |- (A.x x = y <-> A.x(x = x <-> x = y))
76exbii 1049 . 2 |- (E.yA.x x = y <-> E.yA.x(x = x <-> x = y))
8 hbae 1143 . . 3 |- (A.x x = y -> A.yA.x x = y)
9819.9 1034 . 2 |- (E.yA.x x = y <-> A.x x = y)
101, 7, 93bitr2 179 1 |- (E!x x = x <-> A.x x = y)
Colors of variables: wff set class
Syntax hints:   <-> wb 146  A.wal 952   = wceq 954  E.wex 978  E!weu 1378
This theorem is referenced by:  exists2 1456
This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-10 964  ax-12 966  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138
This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 979  df-eu 1380
Copyright terms: Public domain