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Theorem nfiotadxy 4937
Description: Deduction version of nfiotaxy 4938. (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 4935 . 2  |-  ( iota y ps )  = 
U. { z  | 
A. y ( ps  <->  y  =  z ) }
2 nfv 1462 . . . 4  |-  F/ z
ph
3 nfiotadxy.1 . . . . 5  |-  F/ y
ph
4 nfiotadxy.2 . . . . . 6  |-  ( ph  ->  F/ x ps )
5 nfcv 2223 . . . . . . . 8  |-  F/_ x
y
6 nfcv 2223 . . . . . . . 8  |-  F/_ x
z
75, 6nfeq 2230 . . . . . . 7  |-  F/ x  y  =  z
87a1i 9 . . . . . 6  |-  ( ph  ->  F/ x  y  =  z )
94, 8nfbid 1521 . . . . 5  |-  ( ph  ->  F/ x ( ps  <->  y  =  z ) )
103, 9nfald 1685 . . . 4  |-  ( ph  ->  F/ x A. y
( ps  <->  y  =  z ) )
112, 10nfabd 2241 . . 3  |-  ( ph  -> 
F/_ x { z  |  A. y ( ps  <->  y  =  z ) } )
1211nfunid 3634 . 2  |-  ( ph  -> 
F/_ x U. {
z  |  A. y
( ps  <->  y  =  z ) } )
131, 12nfcxfrd 2221 1  |-  ( ph  -> 
F/_ x ( iota y ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103   A.wal 1283    = wceq 1285   F/wnf 1390   {cab 2069   F/_wnfc 2210   U.cuni 3627   iotacio 4932
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-rex 2359  df-sn 3428  df-uni 3628  df-iota 4934
This theorem is referenced by:  nfiotaxy  4938  nfriotadxy  5555
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