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Theorem riota2df 5610
Description: A deduction version of riota2f 5611. (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 270 . . 3  |-  ( (
ph  /\  E! x  e.  A  ps )  ->  B  e.  A )
3 simpr 108 . . . 4  |-  ( (
ph  /\  E! x  e.  A  ps )  ->  E! x  e.  A  ps )
4 df-reu 2366 . . . 4  |-  ( E! x  e.  A  ps  <->  E! x ( x  e.  A  /\  ps )
)
53, 4sylib 120 . . 3  |-  ( (
ph  /\  E! x  e.  A  ps )  ->  E! x ( x  e.  A  /\  ps ) )
6 simpr 108 . . . . . 6  |-  ( ( ( ph  /\  E! x  e.  A  ps )  /\  x  =  B )  ->  x  =  B )
72adantr 270 . . . . . 6  |-  ( ( ( ph  /\  E! x  e.  A  ps )  /\  x  =  B )  ->  B  e.  A )
86, 7eqeltrd 2164 . . . . 5  |-  ( ( ( ph  /\  E! x  e.  A  ps )  /\  x  =  B )  ->  x  e.  A )
98biantrurd 299 . . . 4  |-  ( ( ( ph  /\  E! x  e.  A  ps )  /\  x  =  B )  ->  ( ps  <->  ( x  e.  A  /\  ps ) ) )
10 riota2df.5 . . . . 5  |-  ( (
ph  /\  x  =  B )  ->  ( ps 
<->  ch ) )
1110adantlr 461 . . . 4  |-  ( ( ( ph  /\  E! x  e.  A  ps )  /\  x  =  B )  ->  ( ps  <->  ch ) )
129, 11bitr3d 188 . . 3  |-  ( ( ( ph  /\  E! x  e.  A  ps )  /\  x  =  B )  ->  ( (
x  e.  A  /\  ps )  <->  ch ) )
13 riota2df.1 . . . 4  |-  F/ x ph
14 nfreu1 2538 . . . 4  |-  F/ x E! x  e.  A  ps
1513, 14nfan 1502 . . 3  |-  F/ x
( ph  /\  E! x  e.  A  ps )
16 riota2df.3 . . . 4  |-  ( ph  ->  F/ x ch )
1716adantr 270 . . 3  |-  ( (
ph  /\  E! x  e.  A  ps )  ->  F/ x ch )
18 riota2df.2 . . . 4  |-  ( ph  -> 
F/_ x B )
1918adantr 270 . . 3  |-  ( (
ph  /\  E! x  e.  A  ps )  -> 
F/_ x B )
202, 5, 12, 15, 17, 19iota2df 4991 . 2  |-  ( (
ph  /\  E! x  e.  A  ps )  ->  ( ch  <->  ( iota x ( x  e.  A  /\  ps )
)  =  B ) )
21 df-riota 5590 . . 3  |-  ( iota_ x  e.  A  ps )  =  ( iota x
( x  e.  A  /\  ps ) )
2221eqeq1i 2095 . 2  |-  ( (
iota_ x  e.  A  ps )  =  B  <->  ( iota x ( x  e.  A  /\  ps ) )  =  B )
2320, 22syl6bbr 196 1  |-  ( (
ph  /\  E! x  e.  A  ps )  ->  ( ch  <->  ( iota_ x  e.  A  ps )  =  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1289   F/wnf 1394    e. wcel 1438   E!weu 1948   F/_wnfc 2215   E!wreu 2361   iotacio 4965   iota_crio 5589
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-rex 2365  df-reu 2366  df-v 2621  df-sbc 2839  df-un 3001  df-sn 3447  df-pr 3448  df-uni 3649  df-iota 4967  df-riota 5590
This theorem is referenced by:  riota2f  5611  riota5f  5614
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