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Theorem riota2df 6562
Description: A deduction version of riota2f 6563. (Contributed by NM, 17-Feb-2013.) (Revised by Mario Carneiro, 15-Oct-2016.)
Hypotheses
Ref Expression
riota2df.1  |-  F/ x ph
riota2df.2  |-  ( ph  -> 
F/_ x B )
riota2df.3  |-  ( ph  ->  F/ x ch )
riota2df.4  |-  ( ph  ->  B  e.  A )
riota2df.5  |-  ( (
ph  /\  x  =  B )  ->  ( ps 
<->  ch ) )
Assertion
Ref Expression
riota2df  |-  ( (
ph  /\  E! x  e.  A  ps )  ->  ( ch  <->  ( iota_ x  e.  A ps )  =  B ) )
Distinct variable group:    x, A
Allowed substitution hints:    ph( x)    ps( x)    ch( x)    B( x)

Proof of Theorem riota2df
StepHypRef Expression
1 riota2df.4 . . . 4  |-  ( ph  ->  B  e.  A )
21adantr 452 . . 3  |-  ( (
ph  /\  E! x  e.  A  ps )  ->  B  e.  A )
3 simpr 448 . . . 4  |-  ( (
ph  /\  E! x  e.  A  ps )  ->  E! x  e.  A  ps )
4 df-reu 2704 . . . 4  |-  ( E! x  e.  A  ps  <->  E! x ( x  e.  A  /\  ps )
)
53, 4sylib 189 . . 3  |-  ( (
ph  /\  E! x  e.  A  ps )  ->  E! x ( x  e.  A  /\  ps ) )
6 simpr 448 . . . . . 6  |-  ( ( ( ph  /\  E! x  e.  A  ps )  /\  x  =  B )  ->  x  =  B )
72adantr 452 . . . . . 6  |-  ( ( ( ph  /\  E! x  e.  A  ps )  /\  x  =  B )  ->  B  e.  A )
86, 7eqeltrd 2509 . . . . 5  |-  ( ( ( ph  /\  E! x  e.  A  ps )  /\  x  =  B )  ->  x  e.  A )
98biantrurd 495 . . . 4  |-  ( ( ( ph  /\  E! x  e.  A  ps )  /\  x  =  B )  ->  ( ps  <->  ( x  e.  A  /\  ps ) ) )
10 riota2df.5 . . . . 5  |-  ( (
ph  /\  x  =  B )  ->  ( ps 
<->  ch ) )
1110adantlr 696 . . . 4  |-  ( ( ( ph  /\  E! x  e.  A  ps )  /\  x  =  B )  ->  ( ps  <->  ch ) )
129, 11bitr3d 247 . . 3  |-  ( ( ( ph  /\  E! x  e.  A  ps )  /\  x  =  B )  ->  ( (
x  e.  A  /\  ps )  <->  ch ) )
13 riota2df.1 . . . 4  |-  F/ x ph
14 nfreu1 2870 . . . 4  |-  F/ x E! x  e.  A  ps
1513, 14nfan 1846 . . 3  |-  F/ x
( ph  /\  E! x  e.  A  ps )
16 riota2df.3 . . . 4  |-  ( ph  ->  F/ x ch )
1716adantr 452 . . 3  |-  ( (
ph  /\  E! x  e.  A  ps )  ->  F/ x ch )
18 riota2df.2 . . . 4  |-  ( ph  -> 
F/_ x B )
1918adantr 452 . . 3  |-  ( (
ph  /\  E! x  e.  A  ps )  -> 
F/_ x B )
202, 5, 12, 15, 17, 19iota2df 5434 . 2  |-  ( (
ph  /\  E! x  e.  A  ps )  ->  ( ch  <->  ( iota x ( x  e.  A  /\  ps )
)  =  B ) )
21 riotaiota 6547 . . . 4  |-  ( E! x  e.  A  ps  ->  ( iota_ x  e.  A ps )  =  ( iota x ( x  e.  A  /\  ps )
) )
2221adantl 453 . . 3  |-  ( (
ph  /\  E! x  e.  A  ps )  ->  ( iota_ x  e.  A ps )  =  ( iota x ( x  e.  A  /\  ps )
) )
2322eqeq1d 2443 . 2  |-  ( (
ph  /\  E! x  e.  A  ps )  ->  ( ( iota_ x  e.  A ps )  =  B  <->  ( iota x
( x  e.  A  /\  ps ) )  =  B ) )
2420, 23bitr4d 248 1  |-  ( (
ph  /\  E! x  e.  A  ps )  ->  ( ch  <->  ( iota_ x  e.  A ps )  =  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   F/wnf 1553    = wceq 1652    e. wcel 1725   E!weu 2280   F/_wnfc 2558   E!wreu 2699   iotacio 5408   iota_crio 6534
This theorem is referenced by:  riota2f  6563  riota5f  6566  riotasvdOLD  6585  mapdheq  32463  hdmap1eq  32537  hdmapval2lem  32569
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ral 2702  df-rex 2703  df-reu 2704  df-v 2950  df-sbc 3154  df-un 3317  df-if 3732  df-sn 3812  df-pr 3813  df-uni 4008  df-iota 5410  df-riota 6541
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