ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  eusv1 GIF version

Theorem eusv1 4425
Description: Two ways to express single-valuedness of a class expression 𝐴(𝑥). (Contributed by NM, 14-Oct-2010.)
Assertion
Ref Expression
eusv1 (∃!𝑦𝑥 𝑦 = 𝐴 ↔ ∃𝑦𝑥 𝑦 = 𝐴)
Distinct variable groups:   𝑥,𝑦   𝑦,𝐴
Allowed substitution hint:   𝐴(𝑥)

Proof of Theorem eusv1
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 sp 1498 . . . 4 (∀𝑥 𝑦 = 𝐴𝑦 = 𝐴)
2 sp 1498 . . . 4 (∀𝑥 𝑧 = 𝐴𝑧 = 𝐴)
3 eqtr3 2184 . . . 4 ((𝑦 = 𝐴𝑧 = 𝐴) → 𝑦 = 𝑧)
41, 2, 3syl2an 287 . . 3 ((∀𝑥 𝑦 = 𝐴 ∧ ∀𝑥 𝑧 = 𝐴) → 𝑦 = 𝑧)
54gen2 1437 . 2 𝑦𝑧((∀𝑥 𝑦 = 𝐴 ∧ ∀𝑥 𝑧 = 𝐴) → 𝑦 = 𝑧)
6 eqeq1 2171 . . . 4 (𝑦 = 𝑧 → (𝑦 = 𝐴𝑧 = 𝐴))
76albidv 1811 . . 3 (𝑦 = 𝑧 → (∀𝑥 𝑦 = 𝐴 ↔ ∀𝑥 𝑧 = 𝐴))
87eu4 2075 . 2 (∃!𝑦𝑥 𝑦 = 𝐴 ↔ (∃𝑦𝑥 𝑦 = 𝐴 ∧ ∀𝑦𝑧((∀𝑥 𝑦 = 𝐴 ∧ ∀𝑥 𝑧 = 𝐴) → 𝑦 = 𝑧)))
95, 8mpbiran2 930 1 (∃!𝑦𝑥 𝑦 = 𝐴 ↔ ∃𝑦𝑥 𝑦 = 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  wal 1340   = wceq 1342  wex 1479  ∃!weu 2013
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-ext 2146
This theorem depends on definitions:  df-bi 116  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-cleq 2157
This theorem is referenced by:  eusvnfb  4427
  Copyright terms: Public domain W3C validator