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Theorem exists1 2725
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 5037. (Contributed by NM, 5-Apr-2004.)
Assertion
Ref Expression
exists1 (∃!𝑥 𝑥 = 𝑥 ↔ ∀𝑥 𝑥 = 𝑦)
Distinct variable group:   𝑥,𝑦

Proof of Theorem exists1
StepHypRef Expression
1 df-eu 2636 . 2 (∃!𝑥 𝑥 = 𝑥 ↔ ∃𝑦𝑥(𝑥 = 𝑥𝑥 = 𝑦))
2 equid 2110 . . . . . 6 𝑥 = 𝑥
32tbt 360 . . . . 5 (𝑥 = 𝑦 ↔ (𝑥 = 𝑦𝑥 = 𝑥))
4 bicom 213 . . . . 5 ((𝑥 = 𝑦𝑥 = 𝑥) ↔ (𝑥 = 𝑥𝑥 = 𝑦))
53, 4bitri 266 . . . 4 (𝑥 = 𝑦 ↔ (𝑥 = 𝑥𝑥 = 𝑦))
65albii 1907 . . 3 (∀𝑥 𝑥 = 𝑦 ↔ ∀𝑥(𝑥 = 𝑥𝑥 = 𝑦))
76exbii 1936 . 2 (∃𝑦𝑥 𝑥 = 𝑦 ↔ ∃𝑦𝑥(𝑥 = 𝑥𝑥 = 𝑦))
8 nfae 2482 . . 3 𝑦𝑥 𝑥 = 𝑦
9819.9 2241 . 2 (∃𝑦𝑥 𝑥 = 𝑦 ↔ ∀𝑥 𝑥 = 𝑦)
101, 7, 93bitr2i 290 1 (∃!𝑥 𝑥 = 𝑥 ↔ ∀𝑥 𝑥 = 𝑦)
Colors of variables: wff setvar class
Syntax hints:  wb 197  wal 1635  wex 1859  ∃!weu 2632
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1880  ax-4 1897  ax-5 2004  ax-6 2070  ax-7 2106  ax-10 2187  ax-11 2203  ax-12 2216  ax-13 2422
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-tru 1641  df-ex 1860  df-nf 1865  df-eu 2636
This theorem is referenced by:  exists2  2726
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