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

Theorem nfriotadxy 5616
Description: Deduction version of nfriota 5617. (Contributed by Jim Kingdon, 12-Jan-2019.)
Hypotheses
Ref Expression
nfriotadxy.1  |-  F/ y
ph
nfriotadxy.2  |-  ( ph  ->  F/ x ps )
nfriotadxy.3  |-  ( ph  -> 
F/_ x A )
Assertion
Ref Expression
nfriotadxy  |-  ( ph  -> 
F/_ x ( iota_ y  e.  A  ps )
)
Distinct variable group:    x, y
Allowed substitution hints:    ph( x, y)    ps( x, y)    A( x, y)

Proof of Theorem nfriotadxy
StepHypRef Expression
1 df-riota 5608 . 2  |-  ( iota_ y  e.  A  ps )  =  ( iota y
( y  e.  A  /\  ps ) )
2 nfriotadxy.1 . . 3  |-  F/ y
ph
3 nfcv 2228 . . . . . 6  |-  F/_ x
y
43a1i 9 . . . . 5  |-  ( ph  -> 
F/_ x y )
5 nfriotadxy.3 . . . . 5  |-  ( ph  -> 
F/_ x A )
64, 5nfeld 2244 . . . 4  |-  ( ph  ->  F/ x  y  e.  A )
7 nfriotadxy.2 . . . 4  |-  ( ph  ->  F/ x ps )
86, 7nfand 1505 . . 3  |-  ( ph  ->  F/ x ( y  e.  A  /\  ps ) )
92, 8nfiotadxy 4983 . 2  |-  ( ph  -> 
F/_ x ( iota y ( y  e.  A  /\  ps )
) )
101, 9nfcxfrd 2226 1  |-  ( ph  -> 
F/_ x ( iota_ y  e.  A  ps )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102   F/wnf 1394    e. wcel 1438   F/_wnfc 2215   iotacio 4978   iota_crio 5607
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-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-rex 2365  df-sn 3452  df-uni 3654  df-iota 4980  df-riota 5608
This theorem is referenced by:  nfriota  5617
  Copyright terms: Public domain W3C validator