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Theorem euequ1 2288
 Description: Equality has existential uniqueness. Special case of eueq1 3272 proved using only predicate calculus. The proof needs be free of . This is ensured by having and be distinct. Alternatevly, a distinctor could have been used instead. (Contributed by Stefan Allan, 4-Dec-2008.) (Proof shortened by Wolf Lammen, 8-Sep-2019.)
Assertion
Ref Expression
euequ1
Distinct variable group:   ,

Proof of Theorem euequ1
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 ax6ev 1749 . . 3
2 equequ2 1799 . . . . 5
32equcoms 1795 . . . 4
43alrimiv 1719 . . 3
51, 4eximii 1658 . 2
6 df-eu 2286 . 2
75, 6mpbir 209 1
 Colors of variables: wff setvar class Syntax hints:  <->wb 184  A.wal 1393  E.wex 1612  E!weu 2282 This theorem is referenced by:  copsexg  4737  copsexgOLD  4738  oprabid  6323 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790 This theorem depends on definitions:  df-bi 185  df-ex 1613  df-eu 2286
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