MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  reueq1 Structured version   Unicode version

Theorem reueq1 3053
Description: Equality theorem for restricted uniqueness quantifier. (Contributed by NM, 5-Apr-2004.)
Assertion
Ref Expression
reueq1  |-  ( A  =  B  ->  ( E! x  e.  A  ph  <->  E! x  e.  B  ph ) )
Distinct variable groups:    x, A    x, B
Allowed substitution hint:    ph( x)

Proof of Theorem reueq1
StepHypRef Expression
1 nfcv 2616 . 2  |-  F/_ x A
2 nfcv 2616 . 2  |-  F/_ x B
31, 2reueq1f 3049 1  |-  ( A  =  B  ->  ( E! x  e.  A  ph  <->  E! x  e.  B  ph ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1398   E!wreu 2806
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-an 369  df-tru 1401  df-ex 1618  df-nf 1622  df-eu 2288  df-cleq 2446  df-clel 2449  df-nfc 2604  df-reu 2811
This theorem is referenced by:  reueqd  3061  lubfval  15807  glbfval  15820  isfrgra  25192  frgra3v  25204  1vwmgra  25205  3vfriswmgra  25207  isplig  25381  hdmap14lem4a  37998  hdmap14lem15  38009
  Copyright terms: Public domain W3C validator