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Theorem nfiotad 5222
Description: Deduction version of nfiota 5223. (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 5220 . 2  |-  ( iota y ps )  = 
U. { z  | 
A. y ( ps  <->  y  =  z ) }
2 nfv 1605 . . . 4  |-  F/ z
ph
3 nfiotad.1 . . . . 5  |-  F/ y
ph
4 nfiotad.2 . . . . . . 7  |-  ( ph  ->  F/ x ps )
54adantr 451 . . . . . 6  |-  ( (
ph  /\  -.  A. x  x  =  y )  ->  F/ x ps )
6 nfcvf 2441 . . . . . . . 8  |-  ( -. 
A. x  x  =  y  ->  F/_ x y )
76adantl 452 . . . . . . 7  |-  ( (
ph  /\  -.  A. x  x  =  y )  -> 
F/_ x y )
8 nfcvd 2420 . . . . . . 7  |-  ( (
ph  /\  -.  A. x  x  =  y )  -> 
F/_ x z )
97, 8nfeqd 2433 . . . . . 6  |-  ( (
ph  /\  -.  A. x  x  =  y )  ->  F/ x  y  =  z )
105, 9nfbid 1762 . . . . 5  |-  ( (
ph  /\  -.  A. x  x  =  y )  ->  F/ x ( ps  <->  y  =  z ) )
113, 10nfald2 1912 . . . 4  |-  ( ph  ->  F/ x A. y
( ps  <->  y  =  z ) )
122, 11nfabd 2438 . . 3  |-  ( ph  -> 
F/_ x { z  |  A. y ( ps  <->  y  =  z ) } )
1312nfunid 3834 . 2  |-  ( ph  -> 
F/_ x U. {
z  |  A. y
( ps  <->  y  =  z ) } )
141, 13nfcxfrd 2417 1  |-  ( ph  -> 
F/_ x ( iota y ps ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358   A.wal 1527   F/wnf 1531    = wceq 1623   {cab 2269   F/_wnfc 2406   U.cuni 3827   iotacio 5217
This theorem is referenced by:  nfiota  5223  nfriotad  6313
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-sn 3646  df-uni 3828  df-iota 5219
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