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Theorem nfiotad 5568
Description: Deduction version of nfiota 5569. (Contributed by NM, 18-Feb-2013.)
Hypotheses
Ref Expression
nfiotad.1  |-  F/ y
ph
nfiotad.2  |-  ( ph  ->  F/ x ps )
Assertion
Ref Expression
nfiotad  |-  ( ph  -> 
F/_ x ( iota y ps ) )

Proof of Theorem nfiotad
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 dfiota2 5566 . 2  |-  ( iota y ps )  = 
U. { z  | 
A. y ( ps  <->  y  =  z ) }
2 nfv 1755 . . . 4  |-  F/ z
ph
3 nfiotad.1 . . . . 5  |-  F/ y
ph
4 nfiotad.2 . . . . . . 7  |-  ( ph  ->  F/ x ps )
54adantr 466 . . . . . 6  |-  ( (
ph  /\  -.  A. x  x  =  y )  ->  F/ x ps )
6 nfcvf 2605 . . . . . . . 8  |-  ( -. 
A. x  x  =  y  ->  F/_ x y )
76adantl 467 . . . . . . 7  |-  ( (
ph  /\  -.  A. x  x  =  y )  -> 
F/_ x y )
8 nfcvd 2581 . . . . . . 7  |-  ( (
ph  /\  -.  A. x  x  =  y )  -> 
F/_ x z )
97, 8nfeqd 2587 . . . . . 6  |-  ( (
ph  /\  -.  A. x  x  =  y )  ->  F/ x  y  =  z )
105, 9nfbid 1993 . . . . 5  |-  ( (
ph  /\  -.  A. x  x  =  y )  ->  F/ x ( ps  <->  y  =  z ) )
113, 10nfald2 2132 . . . 4  |-  ( ph  ->  F/ x A. y
( ps  <->  y  =  z ) )
122, 11nfabd 2602 . . 3  |-  ( ph  -> 
F/_ x { z  |  A. y ( ps  <->  y  =  z ) } )
1312nfunid 4226 . 2  |-  ( ph  -> 
F/_ x U. {
z  |  A. y
( ps  <->  y  =  z ) } )
141, 13nfcxfrd 2579 1  |-  ( ph  -> 
F/_ x ( iota y ps ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    /\ wa 370   A.wal 1435   F/wnf 1661   {cab 2407   F/_wnfc 2566   U.cuni 4219   iotacio 5563
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401
This theorem depends on definitions:  df-bi 188  df-an 372  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ral 2776  df-rex 2777  df-sn 3999  df-uni 4220  df-iota 5565
This theorem is referenced by:  nfiota  5569  nfriotad  6276
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