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

Theorem nfriotadxy 5852
Description: Deduction version of nfriota 5853. (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 5844 . 2  |-  ( iota_ y  e.  A  ps )  =  ( iota y
( y  e.  A  /\  ps ) )
2 nfriotadxy.1 . . 3  |-  F/ y
ph
3 nfcv 2329 . . . . . 6  |-  F/_ x
y
43a1i 9 . . . . 5  |-  ( ph  -> 
F/_ x y )
5 nfriotadxy.3 . . . . 5  |-  ( ph  -> 
F/_ x A )
64, 5nfeld 2345 . . . 4  |-  ( ph  ->  F/ x  y  e.  A )
7 nfriotadxy.2 . . . 4  |-  ( ph  ->  F/ x ps )
86, 7nfand 1578 . . 3  |-  ( ph  ->  F/ x ( y  e.  A  /\  ps ) )
92, 8nfiotadw 5193 . 2  |-  ( ph  -> 
F/_ x ( iota y ( y  e.  A  /\  ps )
) )
101, 9nfcxfrd 2327 1  |-  ( ph  -> 
F/_ x ( iota_ y  e.  A  ps )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104   F/wnf 1470    e. wcel 2158   F/_wnfc 2316   iotacio 5188   iota_crio 5843
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169
This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-rex 2471  df-sn 3610  df-uni 3822  df-iota 5190  df-riota 5844
This theorem is referenced by:  nfriota  5853
  Copyright terms: Public domain W3C validator