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Theorem eusv1 5260
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 2181 . . . 4 (∀𝑥 𝑦 = 𝐴𝑦 = 𝐴)
2 sp 2181 . . . 4 (∀𝑥 𝑧 = 𝐴𝑧 = 𝐴)
3 eqtr3 2823 . . . 4 ((𝑦 = 𝐴𝑧 = 𝐴) → 𝑦 = 𝑧)
41, 2, 3syl2an 598 . . 3 ((∀𝑥 𝑦 = 𝐴 ∧ ∀𝑥 𝑧 = 𝐴) → 𝑦 = 𝑧)
54gen2 1798 . 2 𝑦𝑧((∀𝑥 𝑦 = 𝐴 ∧ ∀𝑥 𝑧 = 𝐴) → 𝑦 = 𝑧)
6 eqeq1 2805 . . . 4 (𝑦 = 𝑧 → (𝑦 = 𝐴𝑧 = 𝐴))
76albidv 1921 . . 3 (𝑦 = 𝑧 → (∀𝑥 𝑦 = 𝐴 ↔ ∀𝑥 𝑧 = 𝐴))
87eu4 2679 . 2 (∃!𝑦𝑥 𝑦 = 𝐴 ↔ (∃𝑦𝑥 𝑦 = 𝐴 ∧ ∀𝑦𝑧((∀𝑥 𝑦 = 𝐴 ∧ ∀𝑥 𝑧 = 𝐴) → 𝑦 = 𝑧)))
95, 8mpbiran2 709 1 (∃!𝑦𝑥 𝑦 = 𝐴 ↔ ∃𝑦𝑥 𝑦 = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wal 1536   = wceq 1538  wex 1781  ∃!weu 2631
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-9 2122  ax-12 2176  ax-ext 2773
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-mo 2601  df-eu 2632  df-cleq 2794
This theorem is referenced by:  eusvnfb  5262
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