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

Theorem nfrexya 2498
Description: Not-free for restricted existential quantification where  y and  A are distinct. See nfrexxy 2496 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 1446 . . 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 2493 . 2  |-  ( T. 
->  F/ x E. y  e.  A  ph )
76mptru 1344 1  |-  F/ x E. y  e.  A  ph
Colors of variables: wff set class
Syntax hints:   T. wtru 1336   F/wnf 1440   F/_wnfc 2286   E.wrex 2436
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-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-cleq 2150  df-clel 2153  df-nfc 2288  df-rex 2441
This theorem is referenced by:  nfiunya  3877  nffrec  6343  nfsup  6936  caucvgsrlemgt1  7715  nfsum1  11253  zsupcllemstep  11832  bezout  11895
  Copyright terms: Public domain W3C validator