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Theorem riota2df 6072
Description: A deduction version of riota2f 6073. (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 465 . . 3  |-  ( (
ph  /\  E! x  e.  A  ps )  ->  B  e.  A )
3 simpr 461 . . . 4  |-  ( (
ph  /\  E! x  e.  A  ps )  ->  E! x  e.  A  ps )
4 df-reu 2721 . . . 4  |-  ( E! x  e.  A  ps  <->  E! x ( x  e.  A  /\  ps )
)
53, 4sylib 196 . . 3  |-  ( (
ph  /\  E! x  e.  A  ps )  ->  E! x ( x  e.  A  /\  ps ) )
6 simpr 461 . . . . . 6  |-  ( ( ( ph  /\  E! x  e.  A  ps )  /\  x  =  B )  ->  x  =  B )
72adantr 465 . . . . . 6  |-  ( ( ( ph  /\  E! x  e.  A  ps )  /\  x  =  B )  ->  B  e.  A )
86, 7eqeltrd 2516 . . . . 5  |-  ( ( ( ph  /\  E! x  e.  A  ps )  /\  x  =  B )  ->  x  e.  A )
98biantrurd 508 . . . 4  |-  ( ( ( ph  /\  E! x  e.  A  ps )  /\  x  =  B )  ->  ( ps  <->  ( x  e.  A  /\  ps ) ) )
10 riota2df.5 . . . . 5  |-  ( (
ph  /\  x  =  B )  ->  ( ps 
<->  ch ) )
1110adantlr 714 . . . 4  |-  ( ( ( ph  /\  E! x  e.  A  ps )  /\  x  =  B )  ->  ( ps  <->  ch ) )
129, 11bitr3d 255 . . 3  |-  ( ( ( ph  /\  E! x  e.  A  ps )  /\  x  =  B )  ->  ( (
x  e.  A  /\  ps )  <->  ch ) )
13 riota2df.1 . . . 4  |-  F/ x ph
14 nfreu1 2890 . . . 4  |-  F/ x E! x  e.  A  ps
1513, 14nfan 1861 . . 3  |-  F/ x
( ph  /\  E! x  e.  A  ps )
16 riota2df.3 . . . 4  |-  ( ph  ->  F/ x ch )
1716adantr 465 . . 3  |-  ( (
ph  /\  E! x  e.  A  ps )  ->  F/ x ch )
18 riota2df.2 . . . 4  |-  ( ph  -> 
F/_ x B )
1918adantr 465 . . 3  |-  ( (
ph  /\  E! x  e.  A  ps )  -> 
F/_ x B )
202, 5, 12, 15, 17, 19iota2df 5404 . 2  |-  ( (
ph  /\  E! x  e.  A  ps )  ->  ( ch  <->  ( iota x ( x  e.  A  /\  ps )
)  =  B ) )
21 df-riota 6051 . . 3  |-  ( iota_ x  e.  A  ps )  =  ( iota x
( x  e.  A  /\  ps ) )
2221eqeq1i 2449 . 2  |-  ( (
iota_ x  e.  A  ps )  =  B  <->  ( iota x ( x  e.  A  /\  ps ) )  =  B )
2320, 22syl6bbr 263 1  |-  ( (
ph  /\  E! x  e.  A  ps )  ->  ( ch  <->  ( iota_ x  e.  A  ps )  =  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369   F/wnf 1589    e. wcel 1756   E!weu 2253   F/_wnfc 2565   E!wreu 2716   iotacio 5378   iota_crio 6050
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ral 2719  df-rex 2720  df-reu 2721  df-v 2973  df-sbc 3186  df-un 3332  df-sn 3877  df-pr 3879  df-uni 4091  df-iota 5380  df-riota 6051
This theorem is referenced by:  riota2f  6073  riota5f  6076  mapdheq  35371  hdmap1eq  35445  hdmapval2lem  35477
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