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Theorem eusv2 4135
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 4134 . 2 (∃!yx y = AxA)
3 eusvnfb 4132 . . 3 (∃!yx y = A ↔ (xA A V))
41, 3mpbiran2 834 . 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 1224   = wceq 1226  wex 1358   wcel 1370  ∃!weu 1878  wnfc 2143  Vcvv 2531
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 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000
This theorem depends on definitions:  df-bi 110  df-tru 1229  df-nf 1326  df-sb 1624  df-eu 1881  df-mo 1882  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-rex 2286  df-v 2533  df-sbc 2738  df-csb 2826  df-un 2895  df-sn 3352  df-pr 3353  df-uni 3551
This theorem is referenced by: (None)
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