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

Theorem nfreudxy 2500
Description: Not-free deduction for restricted uniqueness. This is a version where  x and  y are distinct. (Contributed by Jim Kingdon, 6-Jun-2018.)
Hypotheses
Ref Expression
nfreudxy.1  |-  F/ y
ph
nfreudxy.2  |-  ( ph  -> 
F/_ x A )
nfreudxy.3  |-  ( ph  ->  F/ x ps )
Assertion
Ref Expression
nfreudxy  |-  ( ph  ->  F/ x E! y  e.  A  ps )
Distinct variable group:    x, y
Allowed substitution hints:    ph( x, y)    ps( x, y)    A( x, y)

Proof of Theorem nfreudxy
StepHypRef Expression
1 nfreudxy.1 . . 3  |-  F/ y
ph
2 nfcv 2194 . . . . . 6  |-  F/_ x
y
32a1i 9 . . . . 5  |-  ( ph  -> 
F/_ x y )
4 nfreudxy.2 . . . . 5  |-  ( ph  -> 
F/_ x A )
53, 4nfeld 2209 . . . 4  |-  ( ph  ->  F/ x  y  e.  A )
6 nfreudxy.3 . . . 4  |-  ( ph  ->  F/ x ps )
75, 6nfand 1476 . . 3  |-  ( ph  ->  F/ x ( y  e.  A  /\  ps ) )
81, 7nfeud 1932 . 2  |-  ( ph  ->  F/ x E! y ( y  e.  A  /\  ps ) )
9 df-reu 2330 . . 3  |-  ( E! y  e.  A  ps  <->  E! y ( y  e.  A  /\  ps )
)
109nfbii 1378 . 2  |-  ( F/ x E! y  e.  A  ps  <->  F/ x E! y ( y  e.  A  /\  ps )
)
118, 10sylibr 141 1  |-  ( ph  ->  F/ x E! y  e.  A  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101   F/wnf 1365    e. wcel 1409   E!weu 1916   F/_wnfc 2181   E!wreu 2325
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-cleq 2049  df-clel 2052  df-nfc 2183  df-reu 2330
This theorem is referenced by:  nfreuxy  2501
  Copyright terms: Public domain W3C validator