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Theorem eusv1 4184
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 1401 . . . 4 (∀𝑥 𝑦 = 𝐴𝑦 = 𝐴)
2 sp 1401 . . . 4 (∀𝑥 𝑧 = 𝐴𝑧 = 𝐴)
3 eqtr3 2059 . . . 4 ((𝑦 = 𝐴𝑧 = 𝐴) → 𝑦 = 𝑧)
41, 2, 3syl2an 273 . . 3 ((∀𝑥 𝑦 = 𝐴 ∧ ∀𝑥 𝑧 = 𝐴) → 𝑦 = 𝑧)
54gen2 1339 . 2 𝑦𝑧((∀𝑥 𝑦 = 𝐴 ∧ ∀𝑥 𝑧 = 𝐴) → 𝑦 = 𝑧)
6 eqeq1 2046 . . . 4 (𝑦 = 𝑧 → (𝑦 = 𝐴𝑧 = 𝐴))
76albidv 1705 . . 3 (𝑦 = 𝑧 → (∀𝑥 𝑦 = 𝐴 ↔ ∀𝑥 𝑧 = 𝐴))
87eu4 1962 . 2 (∃!𝑦𝑥 𝑦 = 𝐴 ↔ (∃𝑦𝑥 𝑦 = 𝐴 ∧ ∀𝑦𝑧((∀𝑥 𝑦 = 𝐴 ∧ ∀𝑥 𝑧 = 𝐴) → 𝑦 = 𝑧)))
95, 8mpbiran2 848 1 (∃!𝑦𝑥 𝑦 = 𝐴 ↔ ∃𝑦𝑥 𝑦 = 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 97  wb 98  wal 1241   = wceq 1243  wex 1381  ∃!weu 1900
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-cleq 2033
This theorem is referenced by:  eusvnfb  4186
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