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Theorem eueq 3520
Description: Equality has existential uniqueness. (Contributed by NM, 25-Nov-1994.)
Assertion
Ref Expression
eueq (𝐴 ∈ V ↔ ∃!𝑥 𝑥 = 𝐴)
Distinct variable group:   𝑥,𝐴

Proof of Theorem eueq
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 eqtr3 2782 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐴) → 𝑥 = 𝑦)
21gen2 1872 . . 3 𝑥𝑦((𝑥 = 𝐴𝑦 = 𝐴) → 𝑥 = 𝑦)
32biantru 527 . 2 (∃𝑥 𝑥 = 𝐴 ↔ (∃𝑥 𝑥 = 𝐴 ∧ ∀𝑥𝑦((𝑥 = 𝐴𝑦 = 𝐴) → 𝑥 = 𝑦)))
4 isset 3348 . 2 (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
5 eqeq1 2765 . . 3 (𝑥 = 𝑦 → (𝑥 = 𝐴𝑦 = 𝐴))
65eu4 2657 . 2 (∃!𝑥 𝑥 = 𝐴 ↔ (∃𝑥 𝑥 = 𝐴 ∧ ∀𝑥𝑦((𝑥 = 𝐴𝑦 = 𝐴) → 𝑥 = 𝑦)))
73, 4, 63bitr4i 292 1 (𝐴 ∈ V ↔ ∃!𝑥 𝑥 = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  wal 1630   = wceq 1632  wex 1853  wcel 2140  ∃!weu 2608  Vcvv 3341
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-v 3343
This theorem is referenced by:  eueq1  3521  moeq  3524  reuhypd  5045  mptfnf  6177  mptfng  6181  upxp  21649  iotasbc  39141  sprval  42258
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