ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  eusv2 GIF version

Theorem eusv2 4279
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 4278 . 2 (∃!𝑦𝑥 𝑦 = 𝐴𝑥𝐴)
3 eusvnfb 4276 . . 3 (∃!𝑦𝑥 𝑦 = 𝐴 ↔ (𝑥𝐴𝐴 ∈ V))
41, 3mpbiran2 887 . 2 (∃!𝑦𝑥 𝑦 = 𝐴𝑥𝐴)
52, 4bitr4i 185 1 (∃!𝑦𝑥 𝑦 = 𝐴 ↔ ∃!𝑦𝑥 𝑦 = 𝐴)
Colors of variables: wff set class
Syntax hints:  wb 103  wal 1287   = wceq 1289  wex 1426  wcel 1438  ∃!weu 1948  wnfc 2215  Vcvv 2619
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-rex 2365  df-v 2621  df-sbc 2841  df-csb 2934  df-un 3003  df-sn 3452  df-pr 3453  df-uni 3654
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator