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Theorem eusv2i 4153
 Description: Two ways to express single-valuedness of a class expression A(x). (Contributed by NM, 14-Oct-2010.) (Revised by Mario Carneiro, 18-Nov-2016.)
Assertion
Ref Expression
eusv2i (∃!yx y = A∃!yx y = A)
Distinct variable groups:   x,y   y,A
Allowed substitution hint:   A(x)

Proof of Theorem eusv2i
StepHypRef Expression
1 nfeu1 1908 . . 3 y∃!yx y = A
2 nfcvd 2176 . . . . . 6 (∃!yx y = Axy)
3 eusvnf 4151 . . . . . 6 (∃!yx y = AxA)
42, 3nfeqd 2189 . . . . 5 (∃!yx y = A → Ⅎx y = A)
5 nf2 1555 . . . . 5 (Ⅎx y = A ↔ (x y = Ax y = A))
64, 5sylib 127 . . . 4 (∃!yx y = A → (x y = Ax y = A))
7 19.2 1526 . . . 4 (x y = Ax y = A)
86, 7impbid1 130 . . 3 (∃!yx y = A → (x y = Ax y = A))
91, 8eubid 1904 . 2 (∃!yx y = A → (∃!yx y = A∃!yx y = A))
109ibir 166 1 (∃!yx y = A∃!yx y = A)
 Colors of variables: wff set class Syntax hints:   → wi 4  ∀wal 1240   = wceq 1242  Ⅎwnf 1346  ∃wex 1378  ∃!weu 1897 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-v 2553  df-sbc 2759  df-csb 2847 This theorem is referenced by:  eusv2nf  4154
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