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Theorem euequ 2669
 Description: There exists a unique set equal to a given set. Special case of eueqi 3604 proved using only predicate calculus. The proof needs 𝑦 = 𝑧 be free of 𝑥. This is ensured by having 𝑥 and 𝑦 be distinct. Alternately, a distinctor ¬ ∀𝑥𝑥 = 𝑦 could have been used instead. See eueq 3602 and eueqi 3604 for classes. (Contributed by Stefan Allan, 4-Dec-2008.) (Proof shortened by Wolf Lammen, 8-Sep-2019.) Reduce axiom usage. (Revised by Wolf Lammen, 1-Mar-2023.)
Assertion
Ref Expression
euequ ∃!𝑥 𝑥 = 𝑦
Distinct variable group:   𝑥,𝑦

Proof of Theorem euequ
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 ax6ev 2077 . 2 𝑥 𝑥 = 𝑦
2 ax6ev 2077 . . 3 𝑧 𝑧 = 𝑦
3 equeuclr 2127 . . . 4 (𝑧 = 𝑦 → (𝑥 = 𝑦𝑥 = 𝑧))
43alrimiv 2026 . . 3 (𝑧 = 𝑦 → ∀𝑥(𝑥 = 𝑦𝑥 = 𝑧))
52, 4eximii 1935 . 2 𝑧𝑥(𝑥 = 𝑦𝑥 = 𝑧)
6 eu3v 2641 . 2 (∃!𝑥 𝑥 = 𝑦 ↔ (∃𝑥 𝑥 = 𝑦 ∧ ∃𝑧𝑥(𝑥 = 𝑦𝑥 = 𝑧)))
71, 5, 6mpbir2an 702 1 ∃!𝑥 𝑥 = 𝑦
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1654  ∃wex 1878  ∃!weu 2639 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112 This theorem depends on definitions:  df-bi 199  df-an 387  df-ex 1879  df-mo 2605  df-eu 2640 This theorem is referenced by:  copsexg  5178  oprabid  6941
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