ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  eusv2i Structured version   GIF version

Theorem eusv2i 4137
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 1893 . . 3 y∃!yx y = A
2 nfcvd 2161 . . . . . 6 (∃!yx y = Axy)
3 eusvnf 4135 . . . . . 6 (∃!yx y = AxA)
42, 3nfeqd 2174 . . . . 5 (∃!yx y = A → Ⅎx y = A)
5 nf2 1540 . . . . 5 (Ⅎx y = A ↔ (x y = Ax y = A))
64, 5sylib 127 . . . 4 (∃!yx y = A → (x y = Ax y = A))
7 19.2 1511 . . . 4 (x y = Ax y = A)
86, 7impbid1 130 . . 3 (∃!yx y = A → (x y = Ax y = A))
91, 8eubid 1889 . 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 1226   = wceq 1228  wnf 1329  wex 1362  ∃!weu 1882
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 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-v 2537  df-sbc 2742  df-csb 2830
This theorem is referenced by:  eusv2nf  4138
  Copyright terms: Public domain W3C validator