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Theorem eusv2 4155
Description: Two ways to express single-valuedness of a class expression A(x). (Contributed by NM, 15-Oct-2010.) (Proof shortened by Mario Carneiro, 18-Nov-2016.)
Hypothesis
Ref Expression
eusv2.1 A V
Assertion
Ref Expression
eusv2 (∃!yx y = A∃!yx y = A)
Distinct variable groups:   x,y   y,A
Allowed substitution hint:   A(x)

Proof of Theorem eusv2
StepHypRef Expression
1 eusv2.1 . . 3 A V
21eusv2nf 4154 . 2 (∃!yx y = AxA)
3 eusvnfb 4152 . . 3 (∃!yx y = A ↔ (xA A V))
41, 3mpbiran2 847 . 2 (∃!yx y = AxA)
52, 4bitr4i 176 1 (∃!yx y = A∃!yx y = A)
Colors of variables: wff set class
Syntax hints:  wb 98  wal 1240   = wceq 1242  wex 1378   wcel 1390  ∃!weu 1897  wnfc 2162  Vcvv 2551
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-bnd 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-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rex 2306  df-v 2553  df-sbc 2759  df-csb 2847  df-un 2916  df-sn 3373  df-pr 3374  df-uni 3572
This theorem is referenced by: (None)
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