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Theorem riota2df 6341
Description: A deduction version of riota2f 6342. (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 451 . . 3  |-  ( (
ph  /\  E! x  e.  A  ps )  ->  B  e.  A )
3 simpr 447 . . . 4  |-  ( (
ph  /\  E! x  e.  A  ps )  ->  E! x  e.  A  ps )
4 df-reu 2563 . . . 4  |-  ( E! x  e.  A  ps  <->  E! x ( x  e.  A  /\  ps )
)
53, 4sylib 188 . . 3  |-  ( (
ph  /\  E! x  e.  A  ps )  ->  E! x ( x  e.  A  /\  ps ) )
6 simpr 447 . . . . . 6  |-  ( ( ( ph  /\  E! x  e.  A  ps )  /\  x  =  B )  ->  x  =  B )
72adantr 451 . . . . . 6  |-  ( ( ( ph  /\  E! x  e.  A  ps )  /\  x  =  B )  ->  B  e.  A )
86, 7eqeltrd 2370 . . . . 5  |-  ( ( ( ph  /\  E! x  e.  A  ps )  /\  x  =  B )  ->  x  e.  A )
98biantrurd 494 . . . 4  |-  ( ( ( ph  /\  E! x  e.  A  ps )  /\  x  =  B )  ->  ( ps  <->  ( x  e.  A  /\  ps ) ) )
10 riota2df.5 . . . . 5  |-  ( (
ph  /\  x  =  B )  ->  ( ps 
<->  ch ) )
1110adantlr 695 . . . 4  |-  ( ( ( ph  /\  E! x  e.  A  ps )  /\  x  =  B )  ->  ( ps  <->  ch ) )
129, 11bitr3d 246 . . 3  |-  ( ( ( ph  /\  E! x  e.  A  ps )  /\  x  =  B )  ->  ( (
x  e.  A  /\  ps )  <->  ch ) )
13 riota2df.1 . . . 4  |-  F/ x ph
14 nfreu1 2723 . . . 4  |-  F/ x E! x  e.  A  ps
1513, 14nfan 1783 . . 3  |-  F/ x
( ph  /\  E! x  e.  A  ps )
16 riota2df.3 . . . 4  |-  ( ph  ->  F/ x ch )
1716adantr 451 . . 3  |-  ( (
ph  /\  E! x  e.  A  ps )  ->  F/ x ch )
18 riota2df.2 . . . 4  |-  ( ph  -> 
F/_ x B )
1918adantr 451 . . 3  |-  ( (
ph  /\  E! x  e.  A  ps )  -> 
F/_ x B )
202, 5, 12, 15, 17, 19iota2df 5259 . 2  |-  ( (
ph  /\  E! x  e.  A  ps )  ->  ( ch  <->  ( iota x ( x  e.  A  /\  ps )
)  =  B ) )
21 riotaiota 6326 . . . 4  |-  ( E! x  e.  A  ps  ->  ( iota_ x  e.  A ps )  =  ( iota x ( x  e.  A  /\  ps )
) )
2221adantl 452 . . 3  |-  ( (
ph  /\  E! x  e.  A  ps )  ->  ( iota_ x  e.  A ps )  =  ( iota x ( x  e.  A  /\  ps )
) )
2322eqeq1d 2304 . 2  |-  ( (
ph  /\  E! x  e.  A  ps )  ->  ( ( iota_ x  e.  A ps )  =  B  <->  ( iota x
( x  e.  A  /\  ps ) )  =  B ) )
2420, 23bitr4d 247 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 176    /\ wa 358   F/wnf 1534    = wceq 1632    e. wcel 1696   E!weu 2156   F/_wnfc 2419   E!wreu 2558   iotacio 5233   iota_crio 6313
This theorem is referenced by:  riota2f  6342  riota5f  6345  riotasvdOLD  6364  cdlemk36  31724  mapdheq  32540  hdmap1eq  32614  hdmapval2lem  32646
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rex 2562  df-reu 2563  df-v 2803  df-sbc 3005  df-un 3170  df-if 3579  df-sn 3659  df-pr 3660  df-uni 3844  df-iota 5235  df-riota 6320
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