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

Theorem nfrexdxy 2405
Description: Not-free for restricted existential quantification where  x and  y are distinct. See nfrexdya 2407 for a version with  y and  A distinct instead. (Contributed by Jim Kingdon, 30-May-2018.)
Hypotheses
Ref Expression
nfraldxy.2  |-  F/ y
ph
nfraldxy.3  |-  ( ph  -> 
F/_ x A )
nfraldxy.4  |-  ( ph  ->  F/ x ps )
Assertion
Ref Expression
nfrexdxy  |-  ( 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 nfrexdxy
StepHypRef Expression
1 df-rex 2359 . 2  |-  ( E. y  e.  A  ps  <->  E. y ( y  e.  A  /\  ps )
)
2 nfraldxy.2 . . 3  |-  F/ y
ph
3 nfcv 2223 . . . . . 6  |-  F/_ x
y
43a1i 9 . . . . 5  |-  ( ph  -> 
F/_ x y )
5 nfraldxy.3 . . . . 5  |-  ( ph  -> 
F/_ x A )
64, 5nfeld 2238 . . . 4  |-  ( ph  ->  F/ x  y  e.  A )
7 nfraldxy.4 . . . 4  |-  ( ph  ->  F/ x ps )
86, 7nfand 1501 . . 3  |-  ( ph  ->  F/ x ( y  e.  A  /\  ps ) )
92, 8nfexd 1686 . 2  |-  ( ph  ->  F/ x E. y
( y  e.  A  /\  ps ) )
101, 9nfxfrd 1405 1  |-  ( ph  ->  F/ x E. y  e.  A  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102   F/wnf 1390   E.wex 1422    e. wcel 1434   F/_wnfc 2210   E.wrex 2354
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-4 1441  ax-17 1460  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-cleq 2076  df-clel 2079  df-nfc 2212  df-rex 2359
This theorem is referenced by:  nfrexdya  2407  nfrexxy  2409  nfunid  3634  strcollnft  11220
  Copyright terms: Public domain W3C validator