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Theorem nfriotad 6248
Description: Deduction version of nfriota 6249. (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 6240 . 2  |-  ( iota_ y  e.  A  ps )  =  ( iota y
( y  e.  A  /\  ps ) )
2 nfriotad.1 . . . . . 6  |-  F/ y
ph
3 nfnae 2084 . . . . . 6  |-  F/ y  -.  A. x  x  =  y
42, 3nfan 1956 . . . . 5  |-  F/ y ( ph  /\  -.  A. x  x  =  y )
5 nfcvf 2589 . . . . . . . 8  |-  ( -. 
A. x  x  =  y  ->  F/_ x y )
65adantl 464 . . . . . . 7  |-  ( (
ph  /\  -.  A. x  x  =  y )  -> 
F/_ x y )
7 nfriotad.3 . . . . . . . 8  |-  ( ph  -> 
F/_ x A )
87adantr 463 . . . . . . 7  |-  ( (
ph  /\  -.  A. x  x  =  y )  -> 
F/_ x A )
96, 8nfeld 2572 . . . . . 6  |-  ( (
ph  /\  -.  A. x  x  =  y )  ->  F/ x  y  e.  A )
10 nfriotad.2 . . . . . . 7  |-  ( ph  ->  F/ x ps )
1110adantr 463 . . . . . 6  |-  ( (
ph  /\  -.  A. x  x  =  y )  ->  F/ x ps )
129, 11nfand 1953 . . . . 5  |-  ( (
ph  /\  -.  A. x  x  =  y )  ->  F/ x ( y  e.  A  /\  ps ) )
134, 12nfiotad 5536 . . . 4  |-  ( (
ph  /\  -.  A. x  x  =  y )  -> 
F/_ x ( iota y ( y  e.  A  /\  ps )
) )
1413ex 432 . . 3  |-  ( ph  ->  ( -.  A. x  x  =  y  ->  F/_ x ( iota y
( y  e.  A  /\  ps ) ) ) )
15 nfiota1 5535 . . . 4  |-  F/_ y
( iota y ( y  e.  A  /\  ps ) )
16 eqidd 2403 . . . . 5  |-  ( A. x  x  =  y  ->  ( iota y ( y  e.  A  /\  ps ) )  =  ( iota y ( y  e.  A  /\  ps ) ) )
1716drnfc1 2583 . . . 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 2563 1  |-  ( ph  -> 
F/_ x ( iota_ y  e.  A  ps )
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 367   A.wal 1403   F/wnf 1637    e. wcel 1842   F/_wnfc 2550   iotacio 5531   iota_crio 6239
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380
This theorem depends on definitions:  df-bi 185  df-an 369  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ral 2759  df-rex 2760  df-sn 3973  df-uni 4192  df-iota 5533  df-riota 6240
This theorem is referenced by:  nfriota  6249
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