MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  euop2 Unicode version

Theorem euop2 4345
Description: Transfer existential uniqueness to second member of an ordered pair. (Contributed by NM, 10-Apr-2004.)
Hypothesis
Ref Expression
euop2.1  |-  A  e. 
_V
Assertion
Ref Expression
euop2  |-  ( E! x E. y ( x  =  <. A , 
y >.  /\  ph )  <->  E! y ph )
Distinct variable groups:    ph, x    x, A    x, y
Allowed substitution hints:    ph( y)    A( y)

Proof of Theorem euop2
StepHypRef Expression
1 opex 4316 . 2  |-  <. A , 
y >.  e.  _V
2 euop2.1 . . 3  |-  A  e. 
_V
32moop2 4340 . 2  |-  E* y  x  =  <. A , 
y >.
41, 3euxfr2 3026 1  |-  ( E! x E. y ( x  =  <. A , 
y >.  /\  ph )  <->  E! y ph )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358   E.wex 1541    = wceq 1642    e. wcel 1710   E!weu 2209   _Vcvv 2864   <.cop 3719
This theorem is referenced by:  dfac5lem1  7837
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4220  ax-nul 4228  ax-pr 4293
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725
  Copyright terms: Public domain W3C validator