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

Theorem nfrexdxy 2471
Description: Not-free for restricted existential quantification where  x and  y are distinct. See nfrexdya 2473 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 2423 . 2  |-  ( E. y  e.  A  ps  <->  E. y ( y  e.  A  /\  ps )
)
2 nfraldxy.2 . . 3  |-  F/ y
ph
3 nfcv 2282 . . . . . 6  |-  F/_ x
y
43a1i 9 . . . . 5  |-  ( ph  -> 
F/_ x y )
5 nfraldxy.3 . . . . 5  |-  ( ph  -> 
F/_ x A )
64, 5nfeld 2298 . . . 4  |-  ( ph  ->  F/ x  y  e.  A )
7 nfraldxy.4 . . . 4  |-  ( ph  ->  F/ x ps )
86, 7nfand 1548 . . 3  |-  ( ph  ->  F/ x ( y  e.  A  /\  ps ) )
92, 8nfexd 1735 . 2  |-  ( ph  ->  F/ x E. y
( y  e.  A  /\  ps ) )
101, 9nfxfrd 1452 1  |-  ( ph  ->  F/ x E. y  e.  A  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103   F/wnf 1437   E.wex 1469    e. wcel 1481   F/_wnfc 2269   E.wrex 2418
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-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-4 1488  ax-17 1507  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-nf 1438  df-cleq 2133  df-clel 2136  df-nfc 2271  df-rex 2423
This theorem is referenced by:  nfrexdya  2473  nfrexxy  2475  nfunid  3749  strcollnft  13346
  Copyright terms: Public domain W3C validator