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Theorem iota2df 5259
Description: A condition that allows us to represent "the unique element such that  ph " with a class expression  A. (Contributed by NM, 30-Dec-2014.)
Hypotheses
Ref Expression
iota2df.1  |-  ( ph  ->  B  e.  V )
iota2df.2  |-  ( ph  ->  E! x ps )
iota2df.3  |-  ( (
ph  /\  x  =  B )  ->  ( ps 
<->  ch ) )
iota2df.4  |-  F/ x ph
iota2df.5  |-  ( ph  ->  F/ x ch )
iota2df.6  |-  ( ph  -> 
F/_ x B )
Assertion
Ref Expression
iota2df  |-  ( ph  ->  ( ch  <->  ( iota x ps )  =  B ) )

Proof of Theorem iota2df
StepHypRef Expression
1 iota2df.6 . 2  |-  ( ph  -> 
F/_ x B )
2 iota2df.5 . . 3  |-  ( ph  ->  F/ x ch )
3 nfiota1 5237 . . . . 5  |-  F/_ x
( iota x ps )
43a1i 10 . . . 4  |-  ( ph  -> 
F/_ x ( iota
x ps ) )
54, 1nfeqd 2446 . . 3  |-  ( ph  ->  F/ x ( iota
x ps )  =  B )
62, 5nfbid 1774 . 2  |-  ( ph  ->  F/ x ( ch  <->  ( iota x ps )  =  B ) )
7 iota2df.4 . . 3  |-  F/ x ph
8 iota2df.3 . . . . 5  |-  ( (
ph  /\  x  =  B )  ->  ( ps 
<->  ch ) )
9 simpr 447 . . . . . 6  |-  ( (
ph  /\  x  =  B )  ->  x  =  B )
109eqeq2d 2307 . . . . 5  |-  ( (
ph  /\  x  =  B )  ->  (
( iota x ps )  =  x  <->  ( iota x ps )  =  B
) )
118, 10bibi12d 312 . . . 4  |-  ( (
ph  /\  x  =  B )  ->  (
( ps  <->  ( iota x ps )  =  x )  <->  ( ch  <->  ( iota x ps )  =  B ) ) )
1211ex 423 . . 3  |-  ( ph  ->  ( x  =  B  ->  ( ( ps  <->  ( iota x ps )  =  x )  <->  ( ch  <->  ( iota x ps )  =  B ) ) ) )
137, 12alrimi 1757 . 2  |-  ( ph  ->  A. x ( x  =  B  ->  (
( ps  <->  ( iota x ps )  =  x )  <->  ( ch  <->  ( iota x ps )  =  B ) ) ) )
14 iota2df.2 . . . 4  |-  ( ph  ->  E! x ps )
15 iota1 5249 . . . 4  |-  ( E! x ps  ->  ( ps 
<->  ( iota x ps )  =  x ) )
1614, 15syl 15 . . 3  |-  ( ph  ->  ( ps  <->  ( iota x ps )  =  x ) )
177, 16alrimi 1757 . 2  |-  ( ph  ->  A. x ( ps  <->  ( iota x ps )  =  x ) )
18 iota2df.1 . 2  |-  ( ph  ->  B  e.  V )
19 vtoclgft 2847 . 2  |-  ( ( ( F/_ x B  /\  F/ x ( ch  <->  ( iota x ps )  =  B
) )  /\  ( A. x ( x  =  B  ->  ( ( ps 
<->  ( iota x ps )  =  x )  <-> 
( ch  <->  ( iota x ps )  =  B ) ) )  /\  A. x ( ps  <->  ( iota x ps )  =  x ) )  /\  B  e.  V )  ->  ( ch 
<->  ( iota x ps )  =  B ) )
201, 6, 13, 17, 18, 19syl221anc 1193 1  |-  ( ph  ->  ( ch  <->  ( iota x ps )  =  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   A.wal 1530   F/wnf 1534    = wceq 1632    e. wcel 1696   E!weu 2156   F/_wnfc 2419   iotacio 5233
This theorem is referenced by:  iota2d  5260  iota2  5261  opiota  6306  riota2df  6341
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-v 2803  df-sbc 3005  df-un 3170  df-sn 3659  df-pr 3660  df-uni 3844  df-iota 5235
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