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Theorem eusv1 4333
 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 1471 . . . 4 (∀𝑥 𝑦 = 𝐴𝑦 = 𝐴)
2 sp 1471 . . . 4 (∀𝑥 𝑧 = 𝐴𝑧 = 𝐴)
3 eqtr3 2134 . . . 4 ((𝑦 = 𝐴𝑧 = 𝐴) → 𝑦 = 𝑧)
41, 2, 3syl2an 285 . . 3 ((∀𝑥 𝑦 = 𝐴 ∧ ∀𝑥 𝑧 = 𝐴) → 𝑦 = 𝑧)
54gen2 1409 . 2 𝑦𝑧((∀𝑥 𝑦 = 𝐴 ∧ ∀𝑥 𝑧 = 𝐴) → 𝑦 = 𝑧)
6 eqeq1 2121 . . . 4 (𝑦 = 𝑧 → (𝑦 = 𝐴𝑧 = 𝐴))
76albidv 1778 . . 3 (𝑦 = 𝑧 → (∀𝑥 𝑦 = 𝐴 ↔ ∀𝑥 𝑧 = 𝐴))
87eu4 2037 . 2 (∃!𝑦𝑥 𝑦 = 𝐴 ↔ (∃𝑦𝑥 𝑦 = 𝐴 ∧ ∀𝑦𝑧((∀𝑥 𝑦 = 𝐴 ∧ ∀𝑥 𝑧 = 𝐴) → 𝑦 = 𝑧)))
95, 8mpbiran2 908 1 (∃!𝑦𝑥 𝑦 = 𝐴 ↔ ∃𝑦𝑥 𝑦 = 𝐴)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104  ∀wal 1312   = wceq 1314  ∃wex 1451  ∃!weu 1975 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097 This theorem depends on definitions:  df-bi 116  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-cleq 2108 This theorem is referenced by:  eusvnfb  4335
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