MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  eusv4 Unicode version

Theorem eusv4 4543
Description: Two ways to express single-valuedness of a class expression  B ( x ). (Contributed by NM, 27-Oct-2010.)
Hypothesis
Ref Expression
eusv4.1  |-  B  e. 
_V
Assertion
Ref Expression
eusv4  |-  ( E! x E. y  e.  A  x  =  B  <-> 
E! x A. y  e.  A  x  =  B )
Distinct variable groups:    x, y, A    x, B
Allowed substitution hint:    B( y)

Proof of Theorem eusv4
StepHypRef Expression
1 reusv2lem3 4537 . 2  |-  ( A. y  e.  A  B  e.  _V  ->  ( E! x E. y  e.  A  x  =  B  <->  E! x A. y  e.  A  x  =  B )
)
2 eusv4.1 . . 3  |-  B  e. 
_V
32a1i 10 . 2  |-  ( y  e.  A  ->  B  e.  _V )
41, 3mprg 2612 1  |-  ( E! x E. y  e.  A  x  =  B  <-> 
E! x A. y  e.  A  x  =  B )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    = wceq 1623    e. wcel 1684   E!weu 2143   A.wral 2543   E.wrex 2544   _Vcvv 2788
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-nul 4149  ax-pow 4188
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-v 2790  df-dif 3155  df-nul 3456
  Copyright terms: Public domain W3C validator