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Theorem nfriotad 6172
Description: Deduction version of nfriota 6173. (Contributed by NM, 18-Feb-2013.) (Revised by Mario Carneiro, 15-Oct-2016.)
Hypotheses
Ref Expression
nfriotad.1  |-  F/ y
ph
nfriotad.2  |-  ( ph  ->  F/ x ps )
nfriotad.3  |-  ( ph  -> 
F/_ x A )
Assertion
Ref Expression
nfriotad  |-  ( ph  -> 
F/_ x ( iota_ y  e.  A  ps )
)

Proof of Theorem nfriotad
StepHypRef Expression
1 df-riota 6164 . 2  |-  ( iota_ y  e.  A  ps )  =  ( iota y
( y  e.  A  /\  ps ) )
2 nfriotad.1 . . . . . 6  |-  F/ y
ph
3 nfnae 2018 . . . . . 6  |-  F/ y  -.  A. x  x  =  y
42, 3nfan 1866 . . . . 5  |-  F/ y ( ph  /\  -.  A. x  x  =  y )
5 nfcvf 2641 . . . . . . . 8  |-  ( -. 
A. x  x  =  y  ->  F/_ x y )
65adantl 466 . . . . . . 7  |-  ( (
ph  /\  -.  A. x  x  =  y )  -> 
F/_ x y )
7 nfriotad.3 . . . . . . . 8  |-  ( ph  -> 
F/_ x A )
87adantr 465 . . . . . . 7  |-  ( (
ph  /\  -.  A. x  x  =  y )  -> 
F/_ x A )
96, 8nfeld 2624 . . . . . 6  |-  ( (
ph  /\  -.  A. x  x  =  y )  ->  F/ x  y  e.  A )
10 nfriotad.2 . . . . . . 7  |-  ( ph  ->  F/ x ps )
1110adantr 465 . . . . . 6  |-  ( (
ph  /\  -.  A. x  x  =  y )  ->  F/ x ps )
129, 11nfand 1863 . . . . 5  |-  ( (
ph  /\  -.  A. x  x  =  y )  ->  F/ x ( y  e.  A  /\  ps ) )
134, 12nfiotad 5495 . . . 4  |-  ( (
ph  /\  -.  A. x  x  =  y )  -> 
F/_ x ( iota y ( y  e.  A  /\  ps )
) )
1413ex 434 . . 3  |-  ( ph  ->  ( -.  A. x  x  =  y  ->  F/_ x ( iota y
( y  e.  A  /\  ps ) ) ) )
15 nfiota1 5494 . . . 4  |-  F/_ y
( iota y ( y  e.  A  /\  ps ) )
16 eqidd 2455 . . . . 5  |-  ( A. x  x  =  y  ->  ( iota y ( y  e.  A  /\  ps ) )  =  ( iota y ( y  e.  A  /\  ps ) ) )
1716drnfc1 2635 . . . 4  |-  ( A. x  x  =  y  ->  ( F/_ x ( iota y ( y  e.  A  /\  ps ) )  <->  F/_ y ( iota y ( y  e.  A  /\  ps ) ) ) )
1815, 17mpbiri 233 . . 3  |-  ( A. x  x  =  y  -> 
F/_ x ( iota y ( y  e.  A  /\  ps )
) )
1914, 18pm2.61d2 160 . 2  |-  ( ph  -> 
F/_ x ( iota y ( y  e.  A  /\  ps )
) )
201, 19nfcxfrd 2615 1  |-  ( ph  -> 
F/_ x ( iota_ y  e.  A  ps )
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369   A.wal 1368   F/wnf 1590    e. wcel 1758   F/_wnfc 2602   iotacio 5490   iota_crio 6163
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ral 2804  df-rex 2805  df-sn 3989  df-uni 4203  df-iota 5492  df-riota 6164
This theorem is referenced by:  nfriota  6173
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