MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  eusv2 Structured version   Visualization version   GIF version

Theorem eusv2 4856
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 4855 . 2 (∃!𝑦𝑥 𝑦 = 𝐴𝑥𝐴)
3 eusvnfb 4853 . . 3 (∃!𝑦𝑥 𝑦 = 𝐴 ↔ (𝑥𝐴𝐴 ∈ V))
41, 3mpbiran2 953 . 2 (∃!𝑦𝑥 𝑦 = 𝐴𝑥𝐴)
52, 4bitr4i 267 1 (∃!𝑦𝑥 𝑦 = 𝐴 ↔ ∃!𝑦𝑥 𝑦 = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 196  wal 1479   = wceq 1481  wex 1702  wcel 1988  ∃!weu 2468  wnfc 2749  Vcvv 3195
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1484  df-fal 1487  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-nul 3908  df-sn 4169  df-pr 4171  df-uni 4428
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator