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Theorem eusvobj1 5442
 Description: Specify the same object in two ways when class B(y) is single-valued. (Contributed by NM, 1-Nov-2010.) (Proof shortened by Mario Carneiro, 19-Nov-2016.)
Hypothesis
Ref Expression
eusvobj1.1 B V
Assertion
Ref Expression
eusvobj1 (∃!xy A x = B → (℩xy A x = B) = (℩xy A x = B))
Distinct variable groups:   x,y,A   x,B
Allowed substitution hint:   B(y)

Proof of Theorem eusvobj1
StepHypRef Expression
1 nfeu1 1908 . . 3 x∃!xy A x = B
2 eusvobj1.1 . . . 4 B V
32eusvobj2 5441 . . 3 (∃!xy A x = B → (y A x = By A x = B))
41, 3alrimi 1412 . 2 (∃!xy A x = Bx(y A x = By A x = B))
5 iotabi 4819 . 2 (x(y A x = By A x = B) → (℩xy A x = B) = (℩xy A x = B))
64, 5syl 14 1 (∃!xy A x = B → (℩xy A x = B) = (℩xy A x = B))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98  ∀wal 1240   = wceq 1242   ∈ wcel 1390  ∃!weu 1897  ∀wral 2300  ∃wrex 2301  Vcvv 2551  ℩cio 4808 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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-csb 2847  df-sn 3373  df-uni 3572  df-iota 4810 This theorem is referenced by: (None)
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