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Theorem nfiotad 5423
Description: Deduction version of nfiota 5424. (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 5421 . 2  |-  ( iota y ps )  = 
U. { z  | 
A. y ( ps  <->  y  =  z ) }
2 nfv 1630 . . . 4  |-  F/ z
ph
3 nfiotad.1 . . . . 5  |-  F/ y
ph
4 nfiotad.2 . . . . . . 7  |-  ( ph  ->  F/ x ps )
54adantr 453 . . . . . 6  |-  ( (
ph  /\  -.  A. x  x  =  y )  ->  F/ x ps )
6 nfcvf 2596 . . . . . . . 8  |-  ( -. 
A. x  x  =  y  ->  F/_ x y )
76adantl 454 . . . . . . 7  |-  ( (
ph  /\  -.  A. x  x  =  y )  -> 
F/_ x y )
8 nfcvd 2575 . . . . . . 7  |-  ( (
ph  /\  -.  A. x  x  =  y )  -> 
F/_ x z )
97, 8nfeqd 2588 . . . . . 6  |-  ( (
ph  /\  -.  A. x  x  =  y )  ->  F/ x  y  =  z )
105, 9nfbid 1855 . . . . 5  |-  ( (
ph  /\  -.  A. x  x  =  y )  ->  F/ x ( ps  <->  y  =  z ) )
113, 10nfald2 2065 . . . 4  |-  ( ph  ->  F/ x A. y
( ps  <->  y  =  z ) )
122, 11nfabd 2593 . . 3  |-  ( ph  -> 
F/_ x { z  |  A. y ( ps  <->  y  =  z ) } )
1312nfunid 4024 . 2  |-  ( ph  -> 
F/_ x U. {
z  |  A. y
( ps  <->  y  =  z ) } )
141, 13nfcxfrd 2572 1  |-  ( ph  -> 
F/_ x ( iota y ps ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    /\ wa 360   A.wal 1550   F/wnf 1554   {cab 2424   F/_wnfc 2561   U.cuni 4017   iotacio 5418
This theorem is referenced by:  nfiota  5424  nfriotad  6560
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-rex 2713  df-sn 3822  df-uni 4018  df-iota 5420
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