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Theorem eusv1 4265
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 1446 . . . 4 (∀𝑥 𝑦 = 𝐴𝑦 = 𝐴)
2 sp 1446 . . . 4 (∀𝑥 𝑧 = 𝐴𝑧 = 𝐴)
3 eqtr3 2107 . . . 4 ((𝑦 = 𝐴𝑧 = 𝐴) → 𝑦 = 𝑧)
41, 2, 3syl2an 283 . . 3 ((∀𝑥 𝑦 = 𝐴 ∧ ∀𝑥 𝑧 = 𝐴) → 𝑦 = 𝑧)
54gen2 1384 . 2 𝑦𝑧((∀𝑥 𝑦 = 𝐴 ∧ ∀𝑥 𝑧 = 𝐴) → 𝑦 = 𝑧)
6 eqeq1 2094 . . . 4 (𝑦 = 𝑧 → (𝑦 = 𝐴𝑧 = 𝐴))
76albidv 1752 . . 3 (𝑦 = 𝑧 → (∀𝑥 𝑦 = 𝐴 ↔ ∀𝑥 𝑧 = 𝐴))
87eu4 2010 . 2 (∃!𝑦𝑥 𝑦 = 𝐴 ↔ (∃𝑦𝑥 𝑦 = 𝐴 ∧ ∀𝑦𝑧((∀𝑥 𝑦 = 𝐴 ∧ ∀𝑥 𝑧 = 𝐴) → 𝑦 = 𝑧)))
95, 8mpbiran2 887 1 (∃!𝑦𝑥 𝑦 = 𝐴 ↔ ∃𝑦𝑥 𝑦 = 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  wal 1287   = wceq 1289  wex 1426  ∃!weu 1948
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-cleq 2081
This theorem is referenced by:  eusvnfb  4267
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