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

Theorem nfrexxy 2470
Description: Not-free for restricted existential quantification where  x and  y are distinct. See nfrexya 2472 for a version with  y and 
A distinct instead. (Contributed by Jim Kingdon, 30-May-2018.)
Hypotheses
Ref Expression
nfralxy.1  |-  F/_ x A
nfralxy.2  |-  F/ x ph
Assertion
Ref Expression
nfrexxy  |-  F/ x E. y  e.  A  ph
Distinct variable group:    x, y
Allowed substitution hints:    ph( x, y)    A( x, y)

Proof of Theorem nfrexxy
StepHypRef Expression
1 nftru 1442 . . 3  |-  F/ y T.
2 nfralxy.1 . . . 4  |-  F/_ x A
32a1i 9 . . 3  |-  ( T. 
->  F/_ x A )
4 nfralxy.2 . . . 4  |-  F/ x ph
54a1i 9 . . 3  |-  ( T. 
->  F/ x ph )
61, 3, 5nfrexdxy 2466 . 2  |-  ( T. 
->  F/ x E. y  e.  A  ph )
76mptru 1340 1  |-  F/ x E. y  e.  A  ph
Colors of variables: wff set class
Syntax hints:   T. wtru 1332   F/wnf 1436   F/_wnfc 2266   E.wrex 2415
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 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-4 1487  ax-17 1506  ax-ial 1514  ax-i5r 1515  ax-ext 2119
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-cleq 2130  df-clel 2133  df-nfc 2268  df-rex 2420
This theorem is referenced by:  r19.12  2536  sbcrext  2981  nfuni  3737  nfiunxy  3834  rexxpf  4681  abrexex2g  6011  abrexex2  6015  nfrecs  6197  fimaxre2  10991  nfsum  11119  nfcprod1  11316  nfcprod  11317  bezoutlemmain  11675  ctiunctlemfo  11941  bj-findis  13166  strcollnfALT  13173
  Copyright terms: Public domain W3C validator