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Theorem nfiotadxy 4978
Description: Deduction version of nfiotaxy 4979. (Contributed by Jim Kingdon, 21-Dec-2018.)
Hypotheses
Ref Expression
nfiotadxy.1  |-  F/ y
ph
nfiotadxy.2  |-  ( ph  ->  F/ x ps )
Assertion
Ref Expression
nfiotadxy  |-  ( ph  -> 
F/_ x ( iota y ps ) )
Distinct variable group:    x, y
Allowed substitution hints:    ph( x, y)    ps( x, y)

Proof of Theorem nfiotadxy
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 dfiota2 4976 . 2  |-  ( iota y ps )  = 
U. { z  | 
A. y ( ps  <->  y  =  z ) }
2 nfv 1466 . . . 4  |-  F/ z
ph
3 nfiotadxy.1 . . . . 5  |-  F/ y
ph
4 nfiotadxy.2 . . . . . 6  |-  ( ph  ->  F/ x ps )
5 nfcv 2228 . . . . . . . 8  |-  F/_ x
y
6 nfcv 2228 . . . . . . . 8  |-  F/_ x
z
75, 6nfeq 2236 . . . . . . 7  |-  F/ x  y  =  z
87a1i 9 . . . . . 6  |-  ( ph  ->  F/ x  y  =  z )
94, 8nfbid 1525 . . . . 5  |-  ( ph  ->  F/ x ( ps  <->  y  =  z ) )
103, 9nfald 1690 . . . 4  |-  ( ph  ->  F/ x A. y
( ps  <->  y  =  z ) )
112, 10nfabd 2247 . . 3  |-  ( ph  -> 
F/_ x { z  |  A. y ( ps  <->  y  =  z ) } )
1211nfunid 3658 . 2  |-  ( ph  -> 
F/_ x U. {
z  |  A. y
( ps  <->  y  =  z ) } )
131, 12nfcxfrd 2226 1  |-  ( ph  -> 
F/_ x ( iota y ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103   A.wal 1287    = wceq 1289   F/wnf 1394   {cab 2074   F/_wnfc 2215   U.cuni 3651   iotacio 4973
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 3450  df-uni 3652  df-iota 4975
This theorem is referenced by:  nfiotaxy  4979  nfriotadxy  5608
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