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

Theorem nfrexya 2417
Description: Not-free for restricted existential quantification where  y and  A are distinct. See nfrexxy 2415 for a version with  x and  y distinct instead. (Contributed by Jim Kingdon, 3-Jun-2018.)
Hypotheses
Ref Expression
nfralya.1  |-  F/_ x A
nfralya.2  |-  F/ x ph
Assertion
Ref Expression
nfrexya  |-  F/ x E. y  e.  A  ph
Distinct variable group:    y, A
Allowed substitution hints:    ph( x, y)    A( x)

Proof of Theorem nfrexya
StepHypRef Expression
1 nftru 1400 . . 3  |-  F/ y T.
2 nfralya.1 . . . 4  |-  F/_ x A
32a1i 9 . . 3  |-  ( T. 
->  F/_ x A )
4 nfralya.2 . . . 4  |-  F/ x ph
54a1i 9 . . 3  |-  ( T. 
->  F/ x ph )
61, 3, 5nfrexdya 2413 . 2  |-  ( T. 
->  F/ x E. y  e.  A  ph )
76mptru 1298 1  |-  F/ x E. y  e.  A  ph
Colors of variables: wff set class
Syntax hints:   T. wtru 1290   F/wnf 1394   F/_wnfc 2215   E.wrex 2360
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-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-cleq 2081  df-clel 2084  df-nfc 2217  df-rex 2365
This theorem is referenced by:  nfiunya  3753  nffrec  6143  nfsup  6666  caucvgsrlemgt1  7319  nfsum1  10709  zsupcllemstep  11034  bezout  11093
  Copyright terms: Public domain W3C validator