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

Theorem reuss 3325
Description: Transfer uniqueness to a smaller subclass. (Contributed by NM, 21-Aug-1999.)
Assertion
Ref Expression
reuss  |-  ( ( A  C_  B  /\  E. x  e.  A  ph  /\  E! x  e.  B  ph )  ->  E! x  e.  A  ph )
Distinct variable groups:    x, A    x, B
Allowed substitution hint:    ph( x)

Proof of Theorem reuss
StepHypRef Expression
1 idd 21 . . . 4  |-  ( x  e.  A  ->  ( ph  ->  ph ) )
21rgen 2460 . . 3  |-  A. x  e.  A  ( ph  ->  ph )
3 reuss2 3324 . . 3  |-  ( ( ( A  C_  B  /\  A. x  e.  A  ( ph  ->  ph ) )  /\  ( E. x  e.  A  ph  /\  E! x  e.  B  ph )
)  ->  E! x  e.  A  ph )
42, 3mpanl2 429 . 2  |-  ( ( A  C_  B  /\  ( E. x  e.  A  ph 
/\  E! x  e.  B  ph ) )  ->  E! x  e.  A  ph )
543impb 1160 1  |-  ( ( A  C_  B  /\  E. x  e.  A  ph  /\  E! x  e.  B  ph )  ->  E! x  e.  A  ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    /\ w3a 945    e. wcel 1463   A.wral 2391   E.wrex 2392   E!wreu 2393    C_ wss 3039
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-3an 947  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-ral 2396  df-rex 2397  df-reu 2398  df-in 3045  df-ss 3052
This theorem is referenced by:  riotass  5723
  Copyright terms: Public domain W3C validator